Newton-Raphson Procedure Research Articles

BEM applied to damage phenomena in saturated porous media

This paper deals with the numerical analysis of saturated porous media, taking into account the degradation process of the solid skeleton. An implicit boundary element method (BEM) formulation, based on time-independent fundamental solutions, is developed and implemented to couple the fluid flow and the two-dimensional elastostatics problems. The Biot's poro-elastic theory is used and the elastic behavior of the skeleton is coupled to damage. A scalar damage model is assumed for this analysis. The non-linear problem is solved by a Newton-Raphson procedure. A numerical example is presented, in order to validate the implemented formulation and to illustrate its efficiency regarding the accuracy of the results and the robustness of the numerical algorithm.

A two-scale approach for propagating cracks in a fluid-saturated porous material

An extension to a finite strain framework of a two-scale numerical model for propagating crack in porous material is proposed to model the fracture in intervertebral discs. In the model, a crack is described as a propagating cohesive zone by exploiting the partition-of-unity property of finite element shape functions. At the micro-scale, the flow in the cohesive crack is modelled as viscous fluid using Stokes' equations which are averaged over the cross section of the cavity. At the macro-scale, identities are derived to couple the local momentum and the mass balance to the governing equations for a saturated porous material. The resulting discrete equations are nonlinear due to the cohesive constitutive equations and the geometrically nonlinear kinematic relations. A Newton-Raphson iterative procedure is used to consistently linearise the derived system while a Crank-Nicholson scheme takes care of the time integration of the system. The derived model is used to analyse a quasi-static crack growth in confined compression under tensile loading.

Modeling of transport of radionuclides in beds of crushed crystalline rocks under equilibrium non-linear sorption isotherm conditions

Abstract A simple method for fitting the values of the experimental breakthrough curves in the form of pulse response obtained in dynamic flow column experiments is presented. It is based on the equation obtained by the analytical solution of a 1-D advection-dispersion equation (ADE) under defined conditions (equilibrium dynamics, linear sorption isotherm, constant bed height, pulse input), where the concentration (or activity) dependence on the number of pore volumes is expressed explicitly. It is shown that the method can be used in the case of validity of a non-linear Freundlich sorption isotherm if the experimental data are fitted by means of a Newton-Raphson multidimensional non-linear regression procedure in which the regression function consists of the above mentioned ADE equation and of the equation for a retardation coefficient including the first derivative of the Freundlich equation. Values of four parameters, namely, Freundlich equation parameters (nF, kF), Peclet number (Pe) and integration constant (k int) are sought in the course of the regression procedure. Applicability of the method is demonstrated in the evaluation of breakthrough curves of the 137Cs and 85Sr obtained by dynamic flow column experiments in beds of five different types of crushed crystalline rocks, where synthetic groundwater spiked with both radionuclides is used.

The cohesive frictional crack model applied to the analysis of the dam-foundation joint

The mechanical behaviour of dam-foundation joints plays a key role in concrete dam engineering since it is the weakest part of the structure and therefore the evolutionary crack process occurring along this joint determines the global load-bearing capacity. The reference volume involved in the above mentioned process is so large that it cannot be tested in a laboratory: structural analysis has to be carried on by numerical modelling. The use of the asymptotic expansions proposed by Karihaloo and Xiao [13] at the tip of a crack with normal cohesion and Coulomb friction can overcome the numerical difficulties that appear in large scale problems when the Newton–Raphson procedure is applied to a set of equilibrium equations based on ordinary shape functions (Standard Finite Element Method). In this way it is possible to analyze problems with friction and crack propagation under the constant load induced by hydro-mechanical coupling. For each position of the fictitious crack tip, the condition K 1 = K 2 = 0 allows us to obtain the external load level and the tangential stress at the tip. If the joint tangential strength is larger than the value obtained, the solution is acceptable, because the tensile strength is assumed negligible and the condition K 1 = 0 is sufficient to cause the crack growth. Otherwise, the load level obtained can be considered as an overestimation of the critical value and a special form of contact problem has to be solved along the fictitious process zone. For the boundary condition analyzed (ICOLD benchmark on gravity dam model), after an initial increasing phase, the water lag remains almost constant and the maximum value of load carrying capacity is achieved when the water lag reaches its constant value.

Closed form ultimate strength of multi-rectangle reinforced concrete sections under axial load and biaxial bending

The analysis of prismatic members made of reinforced concrete under inclined bending, especially the computation of ultimate loads, is a pronounced non-linear problem which is frequently solved by discretizing the stress distribution in the cross-section using interpolation functions. In the approach described in the present contribution the exact analytical stress distribution is used instead. The obtained expressions are integrated by means of a symbolic manipulation package and automatically converted to optimized Fortran code. The direct problem-computation of ultimate internal forces given the position of the neutral axis-is first described. Subsequently, two kinds of inverse problem are treated: the computation of rupture envelops and the dimensioning of reinforcement, given design internal forces. An iterative Newton-Raphson procedure is used. Examples are presented.

An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection–diffusion equations

In this paper, we present hybridizable discontinuous Galerkin methods for the numerical solution of steady and time-dependent nonlinear convection–diffusion equations. The methods are devised by expressing the approximate scalar variable and corresponding flux in terms of an approximate trace of the scalar variable and then explicitly enforcing the jump condition of the numerical fluxes across the element boundary. Applying the Newton–Raphson procedure and the hybridization technique, we obtain a global equation system solely in terms of the approximate trace of the scalar variable at every Newton iteration. The high number of globally coupled degrees of freedom in the discontinuous Galerkin approximation is therefore significantly reduced. We then extend the method to time-dependent problems by approximating the time derivative by means of backward difference formulae. When the time-marching method is ( p + 1 ) th order accurate and when polynomials of degree p ⩾ 0 are used to represent the scalar variable, each component of the flux and the approximate trace, we observe that the approximations for the scalar variable and the flux converge with the optimal order of p + 1 in the L 2 -norm. Finally, we apply element-by-element postprocessing schemes to obtain new approximations of the flux and the scalar variable. The new approximate flux, which has a continuous interelement normal component, is shown to converge with order p + 1 in the L 2 -norm. The new approximate scalar variable is shown to converge with order p + 2 in the L 2 -norm. The postprocessing is performed at the element level and is thus much less expensive than the solution procedure. For the time-dependent case, the postprocessing does not need to be applied at each time step but only at the times for which an enhanced solution is required. Extensive numerical results are provided to demonstrate the performance of the present method.

Influence of fins on tractor-type podded propulsor performance

A mathematical model of podded propulsors was established in order to investigate the influence of fins. The hydrodynamic performance of podded propulsors with and without fins was calculated, with interactions between propellers and pods and fins derived by iterative calculation. The differential equation based on velocity potential was adopted and hyperboloidal panels were used to avoid gaps between surface panels. The Newton-Raphson iterative procedure was used on the trailing edge to meet the pressure Kutta condition. The velocity distribution was calculated with the Yanagizawa method to eliminate the singularity caused by use of the numerical differential. Comparisons of the performance of podded propulsors with different fins showed that the thrust of propeller in a podded propulsor with fins is greater. The resistance of the pod is also reduced because of the thrust of the fin. The hydrodynamic performance of a podded propulsor with two fins is found to be best, the performance of a podded propulsor with one fin is not as good as two fins, and the performance of the common type is the worst.

A two-loop sparse matrix numerical integration procedure for the solution of differential/algebraic equations: Application to multibody systems

In this paper, a two-loop implicit sparse matrix numerical integration (TLISMNI) procedure for the solution of constrained rigid and flexible multibody system differential and algebraic equations is proposed. The proposed method ensures that the kinematic constraint equations are satisfied at the position, velocity and acceleration levels. In this method, a sparse Lagrangian augmented form of the equations of motion that ensures that the constraints are satisfied at the acceleration level is first used to solve for all the accelerations and Lagrange multipliers. The independent coordinates and velocities are then identified and integrated using HTT or Newmark formulas, expressed in this paper in terms of the independent accelerations only. The constraint equations at the position level are then used within an iterative Newton–Raphson procedure to determine the dependent coordinates. The dependent velocities are determined by solving a linear system of algebraic equations. In order to effectively exploit efficient sparse matrix techniques and have minimum storage requirements, a two-loop iterative method is proposed. Equally important, the proposed method avoids the use of numerical differentiation which is commonly associated with the use of implicit integration methods in multibody system algorithms. Numerical examples are presented in order to demonstrate the use of the new integration procedure.

THREE DIMENSION MODEL FOR SIMULATING INFILTRATION AND REDISTRIBUTION OF FURROW IRRIGATION WATER

Through redistribution, water which entered the soil during infiltration redistributes itself after infiltration has stopped. Both infiltration and redistribution profoundly affect soil water balance. The soil water balance determines the availability of water and nutrients to plants, affects rates of microbial processes, erosion, and chemical weathering, and influence soil thermal and gas composition relations. Therefore, three-dimensional finite difference model for simulating furrow surface flow, infiltration and redistribution water flow under both continuous and surged flow management was developed based on mass balance with the concept of matric flux potential and solved by the Newton-Raphson procedure. Model performance for both continuous and surge flow regimes was verified using field data. Three inflow cycle times ((5/5), (10/10) and (15/15)) were tested with three instantaneous flow rates of 1.45, 1.7 and 2.6 L/s, respectively. Field data were collected to evaluate the advance-recession time for stream flow along furrow length, field infiltration under different flow regimes and soil moisture distribution after irrigation. A sensitivity analysis was made on the response of the model to the changes in specific parameters. Application of the model to surge flow irrigation was demonstrated by analyzing some of the interrelationships between cycle times, flow rates, depth of application efficiency and distribution uniformity. The application efficiency was over 80 % by the surge flow, while it was about 48 % for the continuous flow. Infiltration rate under surge flow approached the basic infiltration in a short time compared to continuous flow. The result showed that a cycle (10/10 min) would create the best distribution uniformity (DU) and application efficiency (AE). The model accurately predicted the transient and steady soil moisture distribution under different inflow and furrow irrigation techniques.

Finite element methods for unsaturated porous solids and their application to dam engineering problems

This work presents a finite element formulation of equations proposed in a companion paper to describe the hyperelastic response of three-phase porous media. Attention is paid to the development of consistent tangents required by the Newton–Raphson procedure used to solve the highly non-linear finite element equations. Among several sources of non-linearity, we also model the permeability dependence on strain as typically observed in intensely jointed rock masses, thus introducing a further reason of hydro-mechanical coupling. Several numerical examples are presented to validate the considered poro-elastic laws and to assess the performance of the numerical formulation. These tests include comparisons with available experimental data relative to a sand column desaturation and with the Philip’s analytical solution for the propagation of a saturation front in an initially dry porous solid. Finally, the formulation is applied to problems of interest for dam engineering, namely the simulation of reservoir bank response to rapid drawdown and a three-dimensional study of concrete gravity dam interaction with foundation and abutment rock masses during reservoir operation.

A Geometric Approach to Maximum Likelihood Estimation of the Functional Principal Components From Sparse Longitudinal Data

In this article, we consider the problem of estimating the eigenvalues and eigenfunctions of the covariance kernel (i.e., the functional principal components) from sparse and irregularly observed longitudinal data. We exploit the smoothness of the eigenfunctions to reduce dimensionality by restricting them to a lower dimensional space of smooth functions. We then approach this problem through a restricted maximum likelihood method. The estimation scheme is based on a Newton–Raphson procedure on the Stiefel manifold using the fact that the basis coefficient matrix for representing the eigenfunctions has orthonormal columns. We also address the selection of the number of basis functions, as well as that of the dimension of the covariance kernel by a second-order approximation to the leave-one-curve-out cross-validation score that is computationally very efficient. The effectiveness of our procedure is demonstrated by simulation studies and an application to a CD4+ counts dataset. In the simulation studies, our method performs well on both estimation and model selection. It also outperforms two existing approaches: one based on a local polynomial smoothing, and another using an EM algorithm. Supplementary materials including technical details, the R package fpca, and data analyzed by this article are available online.

Estimation Of The Parameters Of The Bilinear Seasonal Arima Time Series Model

The parameters of the seasonal bilinear time series model were estimated analytically. This was followed by some numerical illustrations. The closeness of the estimated values to the true values attested to the adequacy of the least square method of minimizing error and the Newton-Raphson iterative procedure. Keywords: Seasonal Bilinear Time Series model, Least Square Method, Newton-Raphson iterative procedure. Global Journal of Mathematical Sciences Vol. 7 (1) 2008: pp. 1-4

Adaptive Mesh Refinement for One-Dimensional Three-Phase Flows in Heterogeneous Fractured Porous Media

This article presents an analysis of the implementation of the adaptive mesh refinement (AMR) technique to one-dimensional three-phase flows in heterogeneous fractured media. Variations of matrix permeability and matrix–fracture transfer shape factor cause matrix saturation discontinuities. In order to avoid directly performing refining or coarsening operations on these matrix quantities, we suggest using two separated grid systems for the fracture and matrix equations, respectively. The fracture and matrix equations are decoupled by a special decomposition approach during the Newton-Raphson procedure for solving the resulting nonlinear equations. The numerical examples show that the proposed AMR technique is fast with good accuracy.

Vibrational Analysis of I2•−.nCO2 Clusters (n = 1−10): A First Principle Study on Microsolvation

Structure, stability, and vibrational IR and Raman spectra of I(2)(*-) x nCO(2) clusters (n = 1-10) are reported based on first-principle electronic structure calculations. Several close-lying minimum energy structures are predicted for these solvated clusters following the quasi Newton-Raphson procedure of geometry optimization. Search strategy based on Monte-Carlo simulated annealing is also applied to find out the global minimum energy structures of these clusters. Successive addition of solvent CO(2) molecules to the negatively charged diatomic solute, I(2)(*-), is fairly symmetrical. Energy parameters of these solvated clusters are calculated following second-order Moller-Plesset perturbation (MP2) as well as coupled cluster theory with 6-311+G(d) set of basis function (I atom is treated with 6-311G(d) set of basis function). The excess electron in these solvated clusters is observed to be localized mainly over the two I atoms. Average interaction energy between the anionic solute, I(2)(*-), and a solvent CO(2) molecule is approximately 129 meV in I(2)(*-) x nCO(2) clusters, and the average interaction energy between two solvent CO(2) molecules is approximately 85 meV in the case of neutral (CO(2))(n) clusters at MP2 level of theory. IR spectra show similar features in all these solvated clusters, depicting a strong band at approximately 2330 cm(-1) for C-O stretching and a weak band at approximately 650 cm(-1) for CO(2) bending modes. Degeneracy of the bending mode of a free solvent CO(2) unit gets lifted when it interacts with the charged solute I(2)(*-) to form a molecular cluster because of the change in structure of solvent CO(2) units. The vibrational band at the bending region of CO(2) in the Raman spectra of these anionic clusters shows a characteristic feature for the formation of I(2)(*-) x nCO(2) clusters showing a Raman band at approximately 650 cm(-1).

Constitutive characterization of rubber blends by means of capillary-viscometry

A new procedure for the material characterization of rubber blends is proposed. It is based on a generalized Newton-Raphson procedure. Such a model has to consider the effect of wall slippage on the viscous properties as well as the non-linear coupling of the viscosity and shear strain rate, which is typical for rubber blends and polymers, respectively. The main goal of this method is the reduction of the experimental effort for determination of the viscosity function. Application of the new method is shown for four rubber materials. An experimental investigation of the viscous properties of rubber blends by means of capillary-viscometry serves as the basis of the work. Because of the pseudo-plastic material behavior of the proposed material, characterization is represented by a coupled system of non-linear equations. Describing their solution requires a numerical integration algorithm. Verification of the developed method for parameter identification was done by comparison of the obtained viscosity curves and the respective velocity profiles in capillary dies with the one obtained by the commonly used material characterization based on correction methods.

IMPROVEMENTS TO THE RADIUS OF CONVERGENCE OF DIRECT INVERSION TECHNIQUES IN HELIUM ATOM SCATTERING

Recent studies (I. G. Shuttleworth, Surf. Rev. Lett.14(2) (2007) 321) have demonstrated a direct method of diffraction pattern inversion for the helium atom scattering (HAS) experiment. The method requires the expansion of the Patterson function as a multivariate Taylor series to form a set of simultaneous equations. Benchmark tests of the procedure show that incomplete Taylor expansions introduce inconsistency into the set of simultaneous equations, whereas larger Taylor expansions attract significant numerical errors during their solution. The current work replaces the conventional matrix inversion techniques with a multidimensional Newton–Raphson algorithm. Tests have shown that the Newton–Raphson procedure removes the Taylor series and numerical limitations from the inversion technique for any realistic surface.

Stability of imperfect stiffened conical shells

A general procedure is developed for stability of stiffened conical shells. It is used for studying the sensitivity behavior with respect to the stiffener configurations. The effect of the pre-buckling nonlinearity on the bifurcation point, as well as the limit-point load level, is examined. The unique algorithm presented by the authors is an extended version of an earlier one, adapted for determination of the limit-point load level of imperfect conical shells. The eigenvalue problem is iteratively solved with respect to the nonlinear equilibrium state up to the bifurcation point or to the limit-point load level. A general symbolic code (using MAPLE) was programmed to create the differential operators based on Donnell’s type shell theory. Then the code uses the Galerkin procedure, the Newton–Raphson procedure, and a finite difference scheme for automatic development of an efficient FORTRAN code which is used for the parametric study.

Evaluation of Two Methods for Determining Soil Gas Permeabilities from Pneumatic Tests

Abstract Soil permeabilities are crucial in the analysis of a soil vapor remediation system. This article presents two methods for determining vertical and horizontal soil permeability (kz and kr) from pneumatic test data. Both the first method, the Newton-Raphson procedure, and the second, the sensitivity analysis method, use the analytical solution developed by Shan et al. (1992) to model a steady-state gas flow to a well partially penetrating the vadose zone. The Newton-Raphson method determines soil permeabilities by using gas pressures collected from two or more observation locations. Calculation examples indicate that the method is efficient and reliable in determination of kr and kz values. This method was compared with the sensitivity analysis technique, which is more flexible in accommodating the number of data points. Both methods give kr and kz values very close to the real solutions. However, calculation examples show that the Newton-Raphson method is more liberal in the selection of initial kr values for the iteration process. It also requires fewer iteration steps to reach a convergence in the cases where the number of data points is less than six.

Consolidation of elastic–plastic saturated porous media by the boundary element method

This paper presents a domain boundary element formulation for inelastic saturated porous media with rate-independent behavior for the solid skeleton. The formulation is then applied to elastic–plastic behavior for the solid. Biot’s consolidation theory, extended to include irreversible phenomena is considered and the direct boundary element technique is used for the numerical solution after time discretization by the implicit Euler backward algorithm. The associated nonlinear algebraic problem is solved by the Newton–Raphson procedure whereby the loading/unloading conditions are fully taken into account and the consistent tangent operator defined. Only domain nodes (nodes defined inside the domain) are used to represent all domain values and the corresponding integrals are computed by using an accurate sub-elementation scheme. The developments are illustrated through the Drucker-Prager elastic–plastic model for the solid skeleton and various examples are analyzed with the proposed algorithms.

Initial and final estimates of the Bilinear seasonal time series model

A particular class of non-linear models which has been found to be useful in many fields is the bilinear models. A special class of it is discussed in this paper. In getting the estimates of the parameters of this model special attention was paid to the problem of having good initial estimates as it is proposed that with good initial values of the parameters the estimates obtaining by the Newton-Raphson iterative technique usually not only converge but also are good estimates. In this paper we examined the initial and final estimates of the bilinear seasonal time series model. The Box-Jenkins linear convergence process, the Newton-Raphson iterative procedure, the Fortran Progran and the MINITAB software package were all employed in achieving both the initial estimates and the final estimates of the bilinear seasonal time series model studied. The results showed considerable closeness between the initial estimates and the final estimates for both simulations (n - 100 and n - 500). This confirmed that the initial estimates are good enough. The implication of this is that in estimations of this nature efforts should be made using the right procedures to achieve good initial estimates so that the final estimates could be achieved quickly after few iterations. JONAMP Vol. 11 2007: pp. 615-620