Predictor Step Research Articles

Variational asymptotic homogenization of finitely deformed heterogeneous elastomers

The objective of this paper is to develop a micromechanics approach to the homogenization and macroscopic stability analysis of finitely deformed heterogeneous elastomers. An Euler–Newton predictor–corrector method is developed for homogenization. It consists of an Euler predictor and a Newton corrector step. Each step involves formulating a variational statement using the variational asymptotic method, discretizing the statement in a finite-dimensional space, and solving the problem using an Euler/multilevel Newton method. An explicit expression for the effective tangent stiffness is obtained in the Euler predictor step and then used for macroscopic stability analysis. The present approach is validated (1) by homogenizing long fiber- and short fiber-reinforced elastomers undergoing uniaxial, biaxial, or shear deformation and (2) by predicting the onset-of-failure curves of porous elastomers with square and hexagonal arrangements of cylindrical voids, undergoing transverse biaxial compression. The present approach is found to be capable of handling various microstructures and complex loading conditions. Different failure modes are found to compete for material failure. More sophisticated hyperelastic material models can be implemented in the present approach.

PEMODELAN KADAR AIR PADA SIFAT FISIK STABILISASI TANAH GAMBUT

: Peat soil is an organic soil with very low carrying capacity and high compressibility. The condition is less profitable for civil engineers in building a civil foundation foundation. One method of peat soil improvement can be done with astabilization method that more environment-friendly and cheaper than other methods. Laboratory based peat stabilization studies to increase carrying capacity, reduce compression and improve peat soil physical properties have been conducted in Indonesia. The results of laboratory studies shown in the graph are still limited by time and content of the mixture. Therefore, further research is needed on the mathematical model toward the physical properties of peat soil stabilization. In this research will be formulated mathematical model of water content on the physical properties of peat soil stabilization. The model is derived from the fluid equation through porous media based on the principle of continum and controlvolume. The model is then resolved numerically by different method until MacCormack scheme with two steps are predictor step using forward difference and correctorstep using backward difference. The MacCormack scheme has the advantage of solving fluid flow equations and continuity. The model is then simulated and validated by comparing the simulation results with the real system. From the simulation results obtained the water content gradually decreased, the decrease is almost close to zero. In addition, it can be seen the difference in decrease in moisture content at each test point although in small quantities. Keywords: Moisture Content, MacCormack, Peat Soil Stabilization.

A robust asymmetrical contact algorithm for explicit solid dynamics

We describe a novel algorithm for the robust approximation of elastic, inelastic, and frictional contact problems in explicit computations. The method is based on a master-slave concept and a predictor/corrector split of the dynamic update. In the predictor step, the bodies move ignoring all contact interactions; in the correction, the nodes that have penetrated a body are pushed back while correcting their velocities to preserve linear momentum and balance the kinetic energy. In contrast with existing predictor/corrector contact algorithms, no iterations nor global computations are required in the correction step. Moreover, thanks to the geometrical basis of the method, the choice of an artificial penalty parameter is avoided. The contact algorithm does not require the computation of the normal vectors at the contacting surfaces, making it especially useful for simulations that employ finite element and certain meshfree discretizations, and for the simulation of contact among bodies with non-smooth boundaries.

A New Method of Integrating the Equations of Autonomous Strapdown INS

We propose the new version of separating the process of integrating the differential equations, which describe the functioning of the strapdown inertial navigation system (SINS) in the normal geographic coordinate system (NGCS), into rapid and slow cycles. In this version, the vector of the relative velocity of an object is represented as a sum of a rapidly changing component and a slowly changing component. The equation for the rapidly changing component of the relative velocity includes the vectors of angular velocities of the Earth’s rotation, NGCS rotation, and, at the same time, the vectors of the apparent acceleration and gravity acceleration, because these accelerations partially balance each other, and at rest relative to the Earth are balanced completely. The equation of the slowly changing component of the relative velocity includes only the vector of angular velocity of the Earth’s rotation and the vector of NGCS rotation. The quaternion orientation of an object relative to the NGCS is represented as a product of two quaternions: a rapidly changing one, which is determined by the absolute angular velocity of an object, and slowly changing one, which is determined by the angular velocity of the NGCS. The right parts of the equations for each group of variables depend on the rapidly changing and slowly changing variables. In order to enable the independent integration of the slow and rapid cycle equations, the algorithm have been developed for integrating the equations using the predictor and corrector for the cases of instantaneous and integral information generated by SINS sensors. At each predictor step the Euler method is used to estimate the longitude, latitude and altitude of an object, slowly changing component of the relative velocity, and slowly changing multiplier of the orientation quaternion at the rightmost point of the slow cycle. Then the Euler-Cauchy method is used to integrate the equations for the rapidly changing components on the rapid cycle intervals, which are present in the slow cycle. The necessary values of the slowly changing components in the intermediate points are calculated using the formulas of linear interpolation. After the rapidly changing components are estimated at the rightmost point of the slow cycle, at the corrector step the Euler-Cauchy method is used to refine the values of the slowly changing components at the rightmost point of the slow cycle. Note that at the beginning of each slow cycle step the slowly changing component of velocity is equal to the value of the relative velocity of an object, and the rapidly changing component is zero. Similarly, at the beginning of each slow cycle step the slowly changing multiplier of object’s orientation quaternion equals to the quaternion of orientation of an object relative to the NGCS, and the rapidly changing multiplier of the orientation of an object has its scalar part equal to one, and its vector part equal to zero (this formula is derived from the quaternion formula for adding the finite rotations). SINS on a stationary base had been simulated in the presence of perturbations for a large time interval for a diving object, which drastically changes its height over short time periods.

The Stability of Incremental Analysis Update.

A recent attempt to downscale the 50 km MERRA-2 analyses to 7 km revealed an instability associated with the Incremental Analysis Update (IAU) procedure that has thus far gone unnoticed. A theoretical study based on a simple damped harmonic oscillator with complex frequency provides the framework to diagnose the problem and suggests means to avoid it. Three possible approaches to avoid the instability are to: (i) choose an "ideal" ratio of the lengths of the Predictor and Corrector steps of IAU based on a theoretical stability diagram; (ii) time average the background fields used to construct the IAU tendencies with given frequency; or (iii) apply a digital filter modulation to the IAU tendencies. All these are shown to control the instability for a wide range of resolutions when doing up- or down-scaling, experiments with the NASA/GMAO atmospheric general circulation model. Furthermore, it is found that combining IAU with the ensemble re-centering step typical of hybrid ensemble-variational approaches, also results in an instability based on the same mechanisms in the members of the ensemble. An example of such occurrence arises in an experiment performed with the GMAO 12.8 km hybrid 4D-EnVar system. Modulation of the ensemble IAU tendencies with a digital filter is shown to avoid the instability. In addition, the stability of certain 4DIAU implementations is analyzed and a suggestion is made to improve its results, though a complete study of this subject is postponed to a follow up work.

The barely implicit correction algorithm for low-Mach-Number flows

A new Barely Implicit Correction (BIC) algorithm is presented for the simulation of low-Mach-number flows. This new algorithm is based on the original, introduced by G. Patnaik et al. [G. Patnaik, R. H. Guirguis, J. P. Boris and E. S. Oran, A barely implicit correction for flux-corrected transport. In: Journal of Computational Physics 71.1 (1987), pp. 120], which was a solution procedure including an explicit predictor step to solve the convective portion of the Navier–Stokes equations and an implicit corrector step to remove the acoustic limit on the integration time-step. The explicit predictor uses a high-order monotone algorithm while the implicit corrector solves an elliptic equation for a pressure correction to equilibrate acoustic waves. In this paper, we develop and extend BIC for multidimensional viscous flows. We introduce a new filter to further stabilize the algorithm and clarify the solution procedure for the inclusion of the viscous fluxes. The new algorithm is examined in three test problems with successively increased difficulty. First, a two-dimensional lid-driven cavity flow is simulated to demonstrate the ability of BIC on solving steady-state swirling flows. Using time steps at least 100 times larger than the explicit limit, good agreements are obtained for solutions when compared with an incompressible calculation by a prior work. A two-dimensional (2D) doubly periodic shear layer flow is simulated to examine the algorithm on solving a transient flow with strong vorticity gradients. Finally, vortex breakdown in three-dimensional (3D) swirling flows are used to further test the stability and performance of the new BIC algorithm. Comparisons of explicit and implicit BIC calculations of both the 2D doubly periodic shear layer and 3D vortex breakdown are presented side by side. They demonstrate that the new BIC algorithm is able to predict accurate and robust solutions using time steps varying from near the explicit stability limit to tens and hundreds of times larger. Excellent agreement is also obtained when compared with results from other algorithms. We discuss our observations of these computations and features which were found to be critical for robustly simulating low-speed, highly dynamic flows.

A new sequential method for three-phase immiscible flow in poroelastic media

A new computational model is developed to solve three phase immiscible compressible flow in strongly heterogeneous poroelastic media. Within the context of the iterative coupled formulation based on the fixed stress split algorithm the governing equations are decomposed into three subsystems associated with the geomechanics, hydrodynamics and transport problems. The hydrodynamics involves an overall compressibility which plays essential role in the magnitude of additional source in the pressure equation associated the time derivative of the total mean stress. We show that the additional coefficients in the source can be decomposed into a rock/grain and fluid components involving the formation volume factors of each phase. The flow equations are written in a proper conservative form for application of mixed-hybrid formulation whereas the nonlinear coupling between flow and transport is handled by a proper sequential iterative scheme. A new post-processing approach is proposed to update the Lagrangian porosity based on the freezing of the total mean stress which reproduces the one-way formulation in a straightforward fashion under the stationarity of the total mean stress. The geomechanical subsystem is also discretized by a mixed formulation based on the decomposition of the effective stress into spherical and deviatoric components in order to handle the nearly-incompressible undrained limit. The system of conservation laws for the liquid saturations is rephrased in an alternative form for an appropriate operator splitting method, leading to the appearance of an extra source term arising from the transient behavior of the Lagrangian porosity induced by the geomechanical coupling. Within the framework of a predictor–corrector scheme, the predictor step is discretized by an extended version of a higher order central-upwind scheme whereas the corrector, which captures the influence of rock skeleton deformation upon transport, is captured by a Godunov splitting. The potential of the new computational model is illustrated in numerical simulations of water-flooding and of WAG (Water Alternating Gas) injection problems in poroelastic media. Numerical results illustrate precisely the role of the transient nature of the total stress upon subsidence, rock compaction, oil and gas production and finger grow in primary and secondary recovery regimes.

Predictor-corrector scheme for simulating wave propagation on shallow water region

In this paper, a well-known Saint-Venant’s Equation is solved by using Predictor-Corrector method to simulate the propagation of wave in shallow water region with several type of bottom friction effects. The predictor step is chosen to be Adam-Bashforth, meanwhile Adam-Moulton for the corrector step. Numerical stability is also discussed by applying von Neumann stability into the linear equations.

Immersed boundary-simplified lattice Boltzmann method for incompressible viscous flows

An immersed boundary-simplified lattice Boltzmann method is developed in this paper for simulations of two-dimensional incompressible viscous flows with immersed objects. Assisted by the fractional step technique, the problem is resolved in a predictor-corrector scheme. The predictor step solves the flow field without considering immersed objects, and the corrector step imposes the effect of immersed boundaries on the velocity field. Different from the previous immersed boundary-lattice Boltzmann method which adopts the standard lattice Boltzmann method (LBM) as the flow solver in the predictor step, a recently developed simplified lattice Boltzmann method (SLBM) is applied in the present method to evaluate intermediate flow variables. Compared to the standard LBM, SLBM requires lower virtual memories, facilitates the implementation of physical boundary conditions, and shows better numerical stability. The boundary condition-enforced immersed boundary method, which accurately ensures no-slip boundary conditions, is implemented as the boundary solver in the corrector step. Four typical numerical examples are presented to demonstrate the stability, the flexibility, and the accuracy of the present method.

A class of efficient high‐order iterative methods with memory for nonlinear equations and their dynamics

In this paper we obtain some theoretical results about iterative methods with memory for nonlinear equations. The class of algorithms we consider focus on incorporating memory without increasing the computational cost of the algorithm. This class uses for the predictor step of each iteration a quantity that has already been calculated in the previous iteration, typically the quantity governing the slope from the previous corrector step. In this way we do not introduce any extra computation, and more importantly, we avoid new function evaluations, allowing us to obtain high‐order iterative methods in a simple way. A specific class of methods of this type is introduced, and we prove the convergence order is 2n+2n−2 with n+1 function evaluations. An exhaustive efficiency study is performed to show the competitiveness of these methods. Finally, we test some specific examples and explore the effect that this predictor may have on the convergence set by setting a dynamical study.

Gross Motor Skills and School Day Physical Activity: Mediating Effect of Perceived Competence

The purpose of this study was to examine the relationship between gross motor skills and school day steps per minute, testing various motivational constructs as potential mediators. A convenience sample of 66 sixth-grade children (mean age = 11.6 ± 0.5 years; 30 boys, 36 girls) were recruited from one public “Zoom” school. Gross motor skills were assessed using the Test for Gross Motor Development-3rd Edition. Motivational constructs were assessed using a series of validated questionnaires. Children wore a pedometer for one school week. A bootstrap mediation analysis was employed using gross motor skills scores as the predictor variable and steps per minute as the outcome variable; the motivational constructs consisted of perceived competence, enjoyment, and self-efficacy as potential mediators. The results from a bootstrap mediation analysis yielded a statistically significant average causal mediation effect (ACME) using perceived competence as the mediator (ACME = 0.022, 95% CI [0.001, 0.054], p = .018). Perceived competence mediated 30.8% of the total effect between gross motor skill scores and steps per minute, with the entire model explaining approximately 13.6% of the variance. The relationship between gross motor skills and school day physical activity may be mediated through perceived competence in sixth-grade children.

An incremental iterative solution procedure without predictor step

Geometric non-linear analyses are performed utilising two main control factors, including load and displacement parameters. In this paper, a new incremental-iterative scheme without predictor step is suggested. Using the obtained constraint equation, the load factor increment is calculated. It is assumed that the path corresponding to the iterative analysis is a parabolic curve. Two different mathematical procedures are developed in order to form this new constraint equation. In the first formulation, two reasonable assumptions are considered for the parabolic path. The length of iterative steps’ curve is minimised in the second scheme. To corroborate the efficiency and capability of the proposed technique, several structures with complex non-linear behaviour are solved.

Fast Sensitivity-Based Nonlinear Economic Model Predictive Control with Degenerate NLP

We present a fast sensitivity-based nonlinear model predictive control (NMPC) algorithm, that can handle non-unique multipliers in the discretized dynamic optimization problem. Non-unique multipliers may arise, for example when path constraints are active for longer periods of the prediction horizon. This is a common situation in economic model predictive control. In such cases, the optimal nonlinear programming (NLP) solution often satisfies the Mangasarian-Fromovitz constraint qualification (MFCQ), which implies non-unique, but bounded multipliers. Consequently, any sensitivity-based fast NMPC scheme must allow for discontinuous jumps in the multipliers. In this paper, we apply a sensitivity-based path-following algorithm that allows multiplier jumps within the advance-step NMPC (asNMPC) framework. The path-following method consists of a corrector and a predictor step, which are computed by solving a system of linear equations, and a quadratic programming problem, respectively, and a multiplier jump step determined by the solution of a linear program. We demonstrate the proposed method on an economic NMPC case study with a CSTR.

Polynomial convergence of two higher order interior-point methods for $$P_*(\kappa )$$ P ∗ ( κ ) -LCP in a wide neighborhood of the central path

In this paper, we propose two interior-point methods for solving \(P_*(\kappa )\)-linear complementarity problems (\(P_*(\kappa )\)-LCPs): a high order large update path following method and a high order corrector–predictor method. Both algorithms generate sequences of iterates in the wide neighborhood \((\mathcal {N}_{2,\tau }^-(\alpha ))\) of the central path introduced by Ai and Zhang. The methods do not depend on the handicap \(\kappa \) of the problem so that they work for any \(P_*(\kappa )\)-LCP . They have \(O((1 +\kappa )\sqrt{n}L)\) iteration complexity, the best-known iteration complexity obtained so far by any interior-point method for solving \(P_*(\kappa )\)-LCP. The high order corrector–predictor algorithm is superlinearly convergent with Q-order \((m_p+1)\) for problems that admit a strict complementarity solution and \((m_p+1)/2\) for general problems, where \(m_p\) is the order of the predictor step.

Phase‐Field Models for the Failure of Anisotropic Continua

AbstractThis study presents a phase‐field approach for an anisotropic continuum to model fracture of biological tissues and fiber‐reinforced composites. We start with the continuous formulation of the variational principle for the multi‐field problem manifested through the deformation map and the crack phase‐field at finite strains which leads to the Euler–Lagrange equations of the coupled problem. In particular, the coupled balance equations derived render the evolution of the anisotropic crack phase‐field and the balance of linear momentum. In addition, we propose a novel energy‐based anisotropic failure criterion which regulates the evolution of the crack phase‐field. The coupled problem is solved using a one‐pass operator‐splitting algorithm composed of a mechanical predictor step and a crack evolution step. Representative numerical examples are devised for crack initiation and propagation in carbon‐fiber‐reinforced polymerg composites. Model parameters are obtained by fitting the set of novel experimental data to the predicted model response; the finite element results qualitatively capture the effect of anisotropy in stiffness and strength. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Wide-Neighborhood Predictor-Corrector Interior-Point Algorithm for Linear Complementarity Problems

In this paper, a wide-neighborhood predictor-corrector feasible interior-point algorithm for linear complementarity problems is proposed. The algorithm is based on using the classical affine scaling direction as a part in a corrector step, not in a predictor step. The convergence analysis of the algorithm is shown, and it is proved that the algorithm has the polynomial complexity $$O\left(\sqrt{n}\log \varepsilon ^{-1}\right)$$ which coincides with the best known iteration bound for this class of mathematical problems. The numerical results indicate the efficiency of the algorithm.

Three-Layer Non-hydrostatic Staggered Scheme for Free Surface Flow

AbstractIn this paper, a finite difference algorithm using a three-layer approximation for the vertical flow region to solve the 2D Euler equations is considered. In this algorithm, the pressure is split into hydrostatic and hydrodynamic parts, and the predictor-corrector procedure is applied. In the predictor step, the momentum hydrostatic model is formulated. In the corrector step, the hydrodynamic pressure is accommodated after solving the Laplace equation using the Successive Over Relaxation (SOR) iteration method. The resulting algorithm is first tested to simulate a standing wave over an intermediate constant depth. Dispersion relation of the scheme is derived, and it is shown to agree with the analytical dispersion relation for kd < π with 94% accuracy. The second test case is a solitary wave simulation. Our computed solitary wave propagates with constant velocity, undisturbed in shape, and confirm the analytical solitary wave. Finally, the scheme is tested to simulate the appearance of the undular bore. The result shows a good agreement with the result from the finite volume scheme for the Boussinesq-type model by Soares-Frazão and Guinot (2008).

An efficient mixed variational reduced‐order model formulation for nonlinear analyses of elastic shells

SummaryThe Koiter‐Newton method had recently demonstrated a superior performance for nonlinear analyses of structures, compared to traditional path‐following strategies. The method follows a predictor‐corrector scheme to trace the entire equilibrium path. During a predictor step, a reduced‐order model is constructed based on Koiter's asymptotic postbuckling theory that is followed by a Newton iteration in the corrector phase to regain the equilibrium of forces. In this manuscript, we introduce a robust mixed solid‐shell formulation to further enhance the efficiency of stability analyses in various aspects. We show that a Hellinger‐Reissner variational formulation facilitates the reduced‐order model construction omitting an expensive evaluation of the inherent fourth‐order derivatives of the strain energy. We demonstrate that extremely large step sizes with a reasonable out‐of‐balance residual can be obtained with substantial impact on the total number of steps needed to trace the complete equilibrium path. More importantly, the numerical effort of the corrector phase involving a Newton iteration of the full‐order model is drastically reduced thus revealing the true strength of the proposed formulation. We study a number of problems from engineering and compare the results to the conventional approach in order to highlight the gain in numerical efficiency for stability problems.

Computational electrodynamics in material media with constraint-preservation, multidimensional Riemann solvers and sub-cell resolution – Part I, second-order FVTD schemes

While classic finite-difference time-domain (FDTD) solutions of Maxwell's equations have served the computational electrodynamics (CED) community very well, formulations based on Godunov methodology have begun to show advantages. We argue that the formulations presented so far are such that FDTD schemes and Godunov-based schemes each have their own unique advantages. However, there is currently not a single formulation that systematically integrates the strengths of both these major strains of development. While an early glimpse of such a formulation was offered in Balsara et al. [16], that paper focused on electrodynamics in plasma. Here, we present a synthesis that integrates the strengths of both FDTD and Godunov-based schemes into a robust single formulation for CED in material media.Three advances make this synthesis possible. First, from the FDTD method, we retain (but somewhat modify) a spatial staggering strategy for the primal variables. This provides a beneficial constraint preservation for the electric displacement and magnetic induction vector fields via reconstruction methods that were initially developed in some of the first author's papers for numerical magnetohydrodynamics (MHD). Second, from the Godunov method, we retain the idea of upwinding, except that this idea, too, has to be significantly modified to use the multi-dimensionally upwinded Riemann solvers developed by the first author. Third, we draw upon recent advances in arbitrary derivatives in space and time (ADER) time-stepping by the first author and his colleagues. We use the ADER predictor step to endow our method with sub-cell resolving capabilities so that the method can be stiffly stable and resolve significant sub-cell variation in the material properties within a zone.Overall, in this paper, we report a new scheme for numerically solving Maxwell's equations in material media, with special attention paid to a second-order-accurate formulation. Several numerical examples are presented to show that the proposed technique works. Because of its sub-cell resolving ability, the new method retains second-order accuracy even when material permeability and permittivity vary by an order-of-magnitude over just one or two zones. Furthermore, because the new method is also unconditionally stable in the presence of stiff source terms (i.e., in problems involving giant conductivity variations), it can handle several orders-of-magnitude variation in material conductivity over just one or two zones without any reduction of the time–step. Consequently, the CFL depends only on the propagation speed of light in the medium being studied.

Primal-dual entropy-based interior-point algorithms for linear optimization

We propose a family of search directions based on primal-dual entropy in the context of interior-point methods for linear optimization. We show that by using entropy-based search directions in the predictor step of a predictor-corrector algorithm together with a homogeneous self-dual embedding, we can achieve the current best iteration complexity bound for linear optimization. Then, we focus on some wide neighborhood algorithms and show that in our family of entropy-based search directions, we can find the best search direction and step size combination by performing a plane search at each iteration. For this purpose, we propose a heuristic plane search algorithm as well as an exact one. Finally, we perform computational experiments to study the performance of entropy-based search directions in wide neighborhoods of the central path, with and without utilizing the plane search algorithms.