An assessment of using the predictor–corrector technique to solve reactive transport equations

We examined, through comparison among the full‐coupling (FC), operator‐splitting (OS), and predictor–corrector (PC) techniques, the effectiveness of using the PC technique to solve depth‐averaged reactive transport equations in the shallow water domain. Our investigation has led to three major conclusions. Firstly, both the OS and PC techniques can efficiently solve reactive transport equations because the advection–diffusion transport equations are solved outside the non‐linear iteration loop and the reaction equations are solved node by node. However, these two techniques may risk sacrificing computational accuracy. Secondly, the OS or PC technique incorporated with the Lagrangian–Eulerian (LE) approach can handle boundary sources more precisely than alternatively with the conventional Eulerian (CE) approach. Thirdly, with the LE approach incorporated, the numerical results from the three techniques agreed highly with one another except when diffusion became significant. In this case, the PC technique's result still matched well with the FC technique's result, but differences between the OS and FC techniques' results arose as diffusion increased. Based on this study, we recommend to apply as a first step the PC technique to solving reactive transport equations with respect to both computational efficiency and accuracy. Copyright © 2002 John Wiley & Sons, Ltd.     

A general conjugate‐gradient‐based predictor–corrector solver for explicit finite‐element contact

A Lagrange‐multiplier based approach is presented for the general solution of multi‐body contact within an explicit finite element framework. The technique employs an explicit predictor step to permit the detection of interpenetration and then utilizes a corrector step, whose solution is obtained with a pre‐conditioned matrix‐free conjugate gradient projection method, to determine the Lagrange multipliers necessary to eliminate the predicted penetration. The predictor–corrector algorithm is developed for deformable bodies based upon the central difference method, and for rigid bodies from momentum and energy conserving approaches. Both frictionless and Coulomb‐based frictional contact idealizations are addressed. The technique imposes no time‐step constraints and quickly mitigates velocity discontinuities across closed interfaces. Special attention is directed toward contact between rigid bodies. Algorithmic moment arms conserve the translational and angular momentums of the system in the absence of external loads. Elastic collisions are captured with a two‐phase predictor–corrector approach and a geometrically approximate velocity jump criterion. The first step solves the inelastic contact problem and identifies inactive constraints between rigid bodies, while the second step generates the necessary velocity jump condition on the active constraints. The velocity criterion is shown to algorithmically preserve the system kinetic energy for two unconstrained rigid bodies. Copyright © 1999 John Wiley & Sons, Ltd. This paper was produced under the auspices of the U.S. Government and it is therefore not subject to copyright in the U.S.     

An alternative approach for numerical solutions of the Navier–Stokes equations

Conventional approaches for solving the Navier–Stokes equations of incompressible fluid dynamics are the primitive‐variable approach and the vorticity–velocity approach. In this paper, an alternative approach is presented. In this approach, pressure and one of the velocity components are eliminated from the governing equations. The result is one higher‐order partial differential equation with one unknown for two‐dimensional problems or two higher‐order partial differential equations with two unknowns for three‐dimensional problems. A meshless collocation method based on radial basis functions for solving the Navier–Stokes equations using this approach is presented. The proposed method is used to solve a two‐ and a three‐dimensional test problem of which exact solutions are known. It is found that, with appropriate values of the method parameters, solutions of satisfactory accuracy can be obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

A combined FIC‐TDG finite element approach for the numerical solution of coupled advection–diffusion–reaction equations with application to a bioregulatory model for bone fracture healing

Numerical schemes for the approximative solution of advection–diffusion–reaction equations are often flawed because of spurious oscillations, caused by steep gradients or dominant advection or reaction. In addition, for strong coupled nonlinear processes, which may be described by a set of hyperbolic PDEs, established time stepping schemes lack either accuracy or stability to provide a reliable solution. In this contribution, an advanced numerical scheme for this class of problems is suggested by combining sophisticated stabilization techniques, namely the finite calculus (FIC‐FEM) scheme introduced by Oñate et al. with time‐discontinuous Galerkin (TDG) methods. Whereas the former one provides a stabilization technique for the numerical treatment of steep gradients for advection‐dominated problems, the latter ensures reliable solutions with regard to the temporal evolution. A brief theoretical outline on the superior behavior of both approaches will be presented and underlined with related computational tests. The performance of the suggested FIC‐TDG finite element approach will be discussed exemplarily on a bioregulatory model for bone fracture healing proposed by Geris et al., which consists of at least 12 coupled hyperbolic evolution equations. Copyright © 2012 John Wiley & Sons, Ltd.     

Physics-data-fusion based decoupling model for coupled faults of complex electromechanical systems

Coupled faults are formed by the nonlinear coupling of multiple lower-level faults in complex electromechanical systems (CES). Although fault decoupling plays a crucial role in locating fault cause and isolating fault components, it still faces challenges due to the harsh reality of common mode failure, networked propagation, and a lack of accurate fault mechanism knowledge in the fault coupling process. A novel physics-data-fusion-based decoupling model for coupled faults of CES was proposed using standard meta components, rigorous formulation, and intuitive representation. First, a hierarchical graph representing the static complex decoupling model was defined by composing proposed meta models. Second, the dynamic model parameters inspired by the time-varying fault characteristics were determined using real-time operation data analysis. Then, based on a proposed numerical reasoning formula, the most likely fault cause was determined, which can also identify fault level by level. Finally, the decoupling model was proved to be reasonable and effective with an offshore wind turbine case. As a graphical modelling method, it handles the decoupling process by fusing static physics and dynamic data of coupled faults. While inheriting the benefits of conventional models, it overcomes the limitations of these existing methods for real-time results. Moreover, the proposed method provided a foundation for tracing the root cause of performance fluctuations, fault detection, and isolation of CES.     

A 3D finite element approach for the coupled numerical simulation of electrochemical systems and fluid flow

A comprehensive finite element method for three‐dimensional simulations of stationary and transient electrochemical systems including all multi‐ion transport mechanisms (convection, diffusion and migration) is presented. In addition, non‐linear phenomenological electrode kinetics boundary conditions are accounted for. The governing equations form a set of coupled non‐linear partial differential equations subject to an algebraic constraint due to the electroneutrality condition. The advantage of a convective formulation of the ion‐transport equations with respect to a natural application of homogeneous flux boundary conditions is emphasized. For one of the numerical examples, an analytical solution for the coupled problem is provided, and it is demonstrated that the proposed computational approach is robust and provides accurate results. Copyright © 2011 John Wiley & Sons, Ltd.     

Substepping schemes for the numerical integration of elastoplastic stress–strain relations

This paper describes two substepping schemes for integrating elastoplastic stress–strain relations. The schemes are designed for use in finite element plasticity calculations and solve for the stress increments assuming that the strain increments are known. Both methods are applicable to a general type of constitutive law and control the error in the integration process by adjusting the size of each substep automatically. The first method is based on the well-known modified Euler scheme, whereas the second technique employs a high order Runge–Kutta formula. The procedures outlined do not require any form of stress correction to prevent drift from the yield surface. Their utility is illustrated by analysis of typical boundary value problems.     

General numerical implementation of a new piezoelectric shunt tuning method based on the effective electromechanical coupling coefficient

Abstract

A recently proposed tuning method for resistive-inductive (RL) shunts is implemented in a commercial finite element (FE) code (ANSYS^®). A main result of the paper is therefore the consistent formulation of the tuning method in terms of variables directly available as solutions in any commercial FE code: The two natural frequencies associated with short- and open-circuit (SC and OC) electrodes and a modal charge obtained as the electrical SC reaction force. An alternative method is based on quasi-static solutions with SC and OC electrodes, convenient for both numerical analysis and experiments. The proposed shunt tuning method is suitable for implementation in any commercial FE software supporting electromechanical analysis and ANSYS^® has been used to assess its accuracy for a piezoelectric smart plate benchmark problem. The method is finally extended to multiple piezoceramic patches, placed symmetrically on the structure and shunted to a single RL network, whereby more vibration modes can be effectively controlled for the specific plate problem.     

A numerical integration approach for fractional-order viscoelastic analysis of hereditary-aging structures

In this paper, a novel numerical integration scheme is proposed for fractional-order viscoelastic analysis of hereditary-aging structures. More precisely, the idea of aging is first introduced through a new phenomenological viscoelastic model characterized by variable-order fractional operators. Then, the presented fractional-order viscoelastic model is included in a variational formulation, conceived for any viscous kernel and discretized in time by employing a discontinuous Galerkin method. The accuracy of the resulting finite element (FE) scheme is analyzed through a model problem, whose exact solution is known; and the most significant variables affecting the solution quality, such as the number of Gaussian quadrature points and time subintervals, are then investigated in terms of error and computational cost. Moreover, the proposed FE integration scheme is applied to study the short- and long-term behavior of concrete structures, which, due to the severe aging exhibited during their service life, represents one of the most challenging time-dependent behavior to be investigated. Eventually, also the Euler implicit method, commonly used in commercial software, is compared.     

A superconducting magnetic levitation device for the transport of light payloads

A device for the frictionless transport of light payloads, e.g., silicon wafers, using superconducting magnetic levitation has been demonstrated. The device consists of an array of rigidly connected carrier magnets levitating above a corresponding array of superconducting discs. Silicon wafers placed on the carrier have been linearly transported with velocities up to 50 cm/s. The configuration provides for excellent lateral stability. Studies of the height and lateral friction effects (caused by flux pinning) were measured as a function of payload mass. Future applications may include the frictionless transport of silicon wafers in vacuum environments.     

A meshfree interpolation method for the numerical solution of the coupled nonlinear partial differential equations

This paper formulates a simple classical radial basis functions (RBFs) collocation (Kansa) method for the numerical solution of the coupled Korteweg-de Vries (KdV) equations, coupled Burgers’ equations, and quasi-nonlinear hyperbolic equations. Contrary to the mesh oriented methods such as the finite-difference and finite element methods, the new technique does not require mesh to discretize the problem domain, and a set of scattered nodes provided by initial data is required for realization of solution of the problem. Accuracy of the method is assessed in terms of the error norms $L_2,L_{\infty }$, number of nodes in the domain of influence, time step length, parameter free and parameter dependent RBFs. Numerical experiments are performed to demonstrate the accuracy and robustness of the method for the three classes of partial differential equations (PDEs).     

A simple spatial integration scheme for solving Cauchy problems of non-linear evolution equations

In this study, we address a new and simple non-iterative method to solve Cauchy problems of non-linear evolution equations without initial data. To start with, these ill-posed problems are analysed by utilizing a semi-discretization numerical scheme. Then, the resulting ordinary differential equations at the discretized times are numerically integrated towards the spatial direction by the group-preserving scheme (GPS). After that, we apply a two-stage GPS to integrate the semi-discretized equations. We reveal that the accuracy and stability of the new approach is very good from several numerical experiments even under a large random noisy effect and a very large time span.     

A block iterative finite element algorithm for numerical solution of the steady–state,compressible Navier–Stokes equations

An iterative method for numerically solving the time independent Navier–Stokes equations for viscous compressible flows is presented. The method is based upon partial application of the Gauss–Seidel principle in block form to the systems of the non-linear algebraic equations which arise in construction of finite element (Galerkin) models approximating solutions of fluid dynamic problems. The C⁰-cubic element on triangles is employed for function approximation. Computational results for a free shear flow at Re = 1000 indicate significant achievement of economy in iterative convergence rate over finite element and finite difference models which employ the customary time dependent equations and symptotic time marching procedure to steady solution. Numerical results are in excellent agreement with those obtained for the same test problem employing time marching finite element and finite difference solution techniques.     

A coupled FE–EFG approach for modelling crack growth in ductile materials

In this work, a coupled finite element–element free Galerkin approach has been used to model crack growth in ductile materials under monotonic and cyclic loads. In this approach, a small discontinuous domain near crack is modelled by EFG method, whereas the rest of the domain is modelled by FEM to exploit the advantages of both the methods. A ramp function has been used in the transition region to maintain the continuity between FE and EFG domains. Two plasticity models (GTN and von‐Mises) and three hardening rules (isotropic, kinematic and mixed) have been used to model the nonlinear material behaviour. Four different problems, i.e. single edge notched tension specimen, double edge notched tension specimen, compact tension specimen and three‐point bend specimen, are solved under plane strain condition using J–R curve approach. Finally, a CT specimen problem is also solved by coupled approach using three hardening rules and two plasticity models under cyclic loading.     

Indirect measurement of state variables for the diagnostics of electromechanical systems

A method of estimating the actual state of electromechanical systems based on the use of a model, connected in parallel with the diagnosed equipment, and an indirect measurement of the state variables is considered. A diagnostic system structure is proposed and also versions of models which take the operating features into account. __________ Translated from Izmeritel’naya Tekhnika, No. 7, pp. 43–46, July, 2007.     

A family of fifth algebraic order trigonometrically fitted Runge–Kutta methods for the numerical solution of the Schrödinger equation

A family of trigonometrically fitted Runge–Kutta methods for the numerical integration of the radial Schrödinger equation is developed. Theoretical and numerical results obtained for the radial Schrödinger equation and for the well known Woods–Saxon potential and for the coupled differential equations of the Schrödinger type show the efficiency of the new method.     

A new hybrid method for solving linear–non‐linear systems of equations

This paper presents a new method for solving any combination of linear–non‐linear equations. The method is based on the separation of linear equations in terms of some selected variables from the non‐linear ones. The linear group is solved by means of any method suitable for the linear system. This operation needs no iteration. The non‐linear group, however, is solved by an iteration technique based on a new formula using the Taylor series expansion. The method has been described and demonstrated in several examples of analytical systems with very good results. The new method needs the initial approximations for non‐linear variables only. This requires far less computation than the Newton–Raphson method. The method also has a very good convergence rate. The proposed method is most beneficial for engineering systems that very often involve a large number of linear equations with limited number of non‐linear equations. Copyright © 2005 John Wiley & Sons, Ltd.     

On approach of crack tip energy release rate for a semi-permeable crack when electromechanical loads become very large

This paper presents an investigation on the approach of the crack tip energy release rate (ERR) for a semi-permeable crack full with air/vacuum or Silicon oil when the electromechanical loads become very large. Numerical results for a central semi-permeable crack, respectively, in seven kinds of piezoelectric ceramics are compared with those for a central impermeable crack when the mechanical loads vary from 50 to 100 MPa and the electric loads are fixed to be 1 MV/m, 0, and –1 MV/m, respectively, within the range of practical interest. It is verified that McMeekings statement (2004): as the electromechanical loads become very large, the crack tip ERR approaches the values associated with an impermeable crack is actually valid under very large mechanical and positive electric loads. However, under very large mechanical and negative electric loads, the approach is quite different showing large discrepancies between the calculated values for the semi-permeable crack and those for an impermeable crack in all seven kinds of piezoelectric ceramics. This means that his statement is not valid when the electric loads are negative even though the mechanical loads still remain very large although, mathematically, McMeekings statement is correct if McMeekings statement: very large is replaced by infinitely large. Moreover, under purely mechanical loads his statement is uncertain, depending on which kind of piezoelectric ceramic is used. It is concluded that, generally speaking, the crack tip ERR for a semi-permeable crack does not approach the values associated with an impermeable crack, depending on the direction of the electric loads with respect to the poling axis. Physically, this is because of the inherent piezoelectric effect that yields the surface charges distributed on the crack surfaces for a semi-permeable crack under the mechanical loads, whereas on the surfaces of an impermeable crack the unphysical charge-free condition leads to incorrect estimations: the applied mechanical loads do not yield any surface charges on the crack surfaces. The influence of the permittivity of medium inside the semi-permeable crack gap on McMeekings statement is discussed too. It is found that Silicon oil yields larger discrepancies than air from those for an impermeable crack.