Semi-implicit Time Integration Scheme Research Articles

Time-domain, spherical harmonic-finite element approach to transient three-dimensional geomagnetic induction in a spherical heterogeneous Earth

SUMMARY We introduce a time-domain method to solve the problem of geomagnetic induction in a heterogeneous Earth excited by variations of the ionospheric and magnetospheric currents with arbitrary spatiotemporal characteristics. The problem is formulated in the integral sense. The Galerkin discretization is based on the vector spherical harmonic functions and piecewise linear finite elements in the angular and radial directions, respectively. A semi-implicit time integration scheme is employed. The method is validated against semi-analytical (SA) solutions for conductivity models consisting of multiple eccentrically nested homogeneous spheres.

Modelling of wetting and drying of shallow water using artificial porosity

AbstractA new method for wetting and drying in two‐dimensional shallow water flow models is proposed. The method is closely related to the artificial porosity method used by different authors in Boussinesq‐type models, but is further extended for use in a semi‐implicit (ADI‐type) time integration scheme. The method is implemented in the simulation model WAQUA using general boundary fitted coordinates and is applied to realistic schematization for a portion of the river Meuse in the Netherlands.A large advantage of the artificial porosity method over traditionally used methods on the basis of ‘screens’ is a strongly reduced sensitivity of model results. Instead of blocking all water transport in grid points where the water level becomes small, as in screen‐based methods, the flow is gradually closed off. Small changes in parameters such as the initial conditions or bottom topography therefore no longer lead to large changes in the model results. Copyright © 2005 John Wiley & Sons, Ltd.

Time Integration Algorithm for a Cyclic Damage Coupled Thermo-Viscoplasticity Model for 63Sn-37Pb Solder Applications

In this paper, a semi-implicit time integration scheme has been developed for a damage-coupled constitutive model to characterize the mechanical behavior of 63Sn-37Pb solder material under thermo-mechanical fatigue (TMF) loading. The scheme is developed to provide an efficient numerical procedure of integration and iteration for calculating stress and other associated state variables within a strain-driven format. In particular, a novel Newton-Raphson iteration algorithm for the damage coupled constitutive material model involving von Mises viscoplastic potential function with nonlinear mixed hardening is formulated. An algorithmic tangent stiffness tensor is derived and the model is implemented numerically into a commercial finite element (FE) code ABAQUS through its user-defined material subroutine. Several numerical simulations are conducted for validation of the proposed algorithm.

Thermally activated magnetization rotation in small nanoparticles

In this paper, we test the performance and validity of a semi-implicit time-integration scheme, originally applied in quantum dynamics, for use in micromagnetics. The attempt frequency and energy barriers are calculated for Co nanoparticles. Moreover, a fit of the relaxation time to the Arrhenius-Neel law is presented. The semi-implicit time-integration method is very robust and allows time steps up to 3 ps at a temperature of 50 K.

Semi-Implicit Spectral Element Atmospheric Model

A semi-implicit spectral element shallow water model on the cubed-sphere is described and numerical results are compared with an explicit formulation and the traditional spectral transform method using the standard test cases proposed by Williamson et al. (1992). The explicit time step is limited by the phase speed of the fastest gravity waves. Semi-implicit time integration schemes remove this stability restriction but require the solution of an elliptic problem. A weak variational formulation of the governing equations leads to a symmetric Helmholtz operator. The resulting implicit problem for the geopotential is then solved using a block-Jacobi preconditioned conjugate gradient solver. The simulation rate of the semi-implicit model is accelerated relative to the explicit model for practical climate resolutions by a factor of two.

Stability of Pressure Boundary Conditions for Stokes and Navier–Stokes Equations

The stability of a finite difference discretization of the time-dependent incompressible Navier–Stokes equations in velocity-pressure formulation is studied. In paticular, we compare the stability for different pressure boundary conditions in a semiimplicit time-integration scheme. where only the viscous term is treated implicitly. The stability is studied in three different ways: by a normal-mode analysis, by numerical computation of the amplification factors, and by direct numerical simulation of the governing equations. All three approaches identify the same pressure boundary condition as the best alternative. This condition implicitly enforces the normal derivative of the divergence to be zero on the boundary by coupling the normal derivative of the pressure to the normal component of the curl of the vorticity. Using this boundary condition, we demonstrate that the time-step is determined only by the convective term.

Low-Reynolds-number effects derived from direct numerical simulations of turbulent pipe flow

Fully developed, statistically steady and non-swirling turbulent flow through straight pipes of circular cross-section is investigated by means of direct numerical simulation. A finite-volume scheme of second-order accuracy and a semi-implicit time integration scheme of the same order of accuracy are used to integrate the incompressible Navier–Stokes equations on staggered grids. Significant low-Reynolds- number effects are observed in the mean axial velocity, the components of the Reynolds stress tensor, the pressure and vorticity fluctuations, the turbulent kinetic energy budget and two-point velocity correlations. It is confirmed that turbulence data obtained in the near-wall region do not scale on wall variables. Within the range of Reynolds numbers investigated, the non-dimensional streak spacing increases slightly with Re τ .

Second‐order accuracy of two‐time‐level semi‐Lagrangian schemes

AbstractThe accuracy of the two‐time‐level semi‐Lagrangian semi‐implicit time integration scheme is analysed. Two relatively independent problems are identified–the discretization of the implicit trajectory equation and the treatment of the nonlinear residual. Two theorems giving conditions for second‐order accuracy of the two‐time‐level semi‐Lagrangian semi‐implicit scheme are stated. As a result, a new scheme for the trajectory equation is proposed and a class of schemes for the nonlinear residual, which are second‐order accurate in time, are introduced. A simplified one‐dimensional ‘shallow water’ model is developed in order to test the proposed scheme in comparison with other, previously known, ones. Some experiments proving stability, accuracy and conservation ability are presented.

Hydrodynamic interactions in long chain polymers: Application of the Chebyshev polynomial approximation in stochastic simulations

We have simulated Brownian bead-spring chains of up to 125 units with fluctuating hydrodynamic and excluded volume interactions using the Chebyshev polynomial approximation proposed by Fixman [Macromolecules 19, 1204 (1986)] for the square root of the diffusion tensor. We have developed a fast method to continuously determine the validity of the eigenvalue range used in the polynomial approximation, and demonstrated how this range may be quickly updated when necessary. We have also developed a weak first order semiimplicit time integration scheme which offers increased stability in the presence of steep excluded volume potentials. The full algorithm scales roughly as O(N2.25) and offers substantial computational savings over the standard Cholesky decomposition. The above algorithm was used to obtain scaling exponents for various static and zero shear rate dynamical properties, which are found to be consistent with theoretical and/or experimental predictions.

The Role of Time Step Size in Numerical Stability of Tangent Linear Models

Abstract It is found that some stable time-integration schemes for some nonlinear models do not guarantee stable integrations of the associated tangent linear models with the same time step size. These problems usually occur when the nonlinear models describe vertical diffusion processes and are numerically implemented by semi-implicit time-integration schemes that are unconditionally stable. The direct linearization procedure performed on such numerical schemes of nonlinear models can be interpreted as some conditionally stable numerical schemes of the underlying linearized equations. Numerical experiments using a simple, illustrative model and a realistic ocean mixed layer model and their tangent linear models showed instabilities in the tangent linear models. Several methods are tried to reduce the nonphysical noise caused by the numerical instabilities. This study suggests that reducing time step size can give good results compared to some other methods that either are not accurate enough or change to...

Parallel Semi-Implicit Spectral Element Methods for Atmospheric General Circulation Models

Spectral elements combine the accuracy and exponential convergence of conventional spectral methods with the geometric flexibility of finite elements. Additionally, there are several apparent computational advantages to using spectral element methods on microprocessors. In particular, the computations are naturally cache-blocked and derivatives may be computed using nearest neighbor communications. Thus, an explicit spectral element atmospheric model has demonstrated close to linear scaling on a variety of distributed memory computers including the IBM SP and Linux Clusters. Explicit formulations of PDE's arising in geophysical fluid dynamics, such as the primitive equations on the sphere, are time-step limited by the phase speed of gravity waves. Semi-implicit time integration schemes remove the stability restriction but require the solution of an elliptic BVP. By employing a weak formulation of the governing equations, it is possible to obtain a symmetric Helmholtz operator that permits the solution of the implicit problem using conjugate gradients. We find that a block-Jacobi preconditioned conjugate gradient solver accelerates the simulation rate of the semi-implicit relative to the explicit formulation for practical climate resolutions by about a factor of three.

Implicit and semi-implicit schemes: Algorithms

This study formulates general guidelines to extend an explicit code with a great variety of implicit and semi-implicit time integration schemes. The discussion is based on their specific implementation in the Versatile Advection Code, which is a general purpose software package for solving systems of non-linear hyperbolic (and/or parabolic) partial differential equations, using standard high resolution shock capturing schemes. For all combinations of explicit high resolution schemes with implicit and semi-implicit treatments, it is shown how second-order spatial and temporal accuracy for the smooth part of the solutions can be maintained. Strategies to obtain steady state and time accurate solutions implicitly are discussed. The implicit and semi-implicit schemes require the solution of large linear systems containing the Jacobian matrix. The Jacobian matrix itself is calculated numerically to ensure the generality of this implementation. Three options are discussed in terms of applicability, storage requirements and computational efficiency. One option is the easily implemented matrix-free approach, but the Jacobian matrix can also be calculated by using a general grid masking algorithm, or by an efficient implementation for a specific Lax–Friedrich-type total variation diminishing (TVD) spatial discretization. The choice of the linear solver depends on the dimensionality of the problem. In one dimension, a direct block tridiagonal solver can be applied, while in more than one spatial dimension, a conjugate gradient (CG)-type iterative solver is used. For advection-dominated problems, preconditioning is needed to accelerate the convergence of the iterative schemes. The modified block incomplete LU-preconditioner is implemented, which performs very well. Examples from two-dimensional hydrodynamic and magnetohydrodynamic computations are given. They model transonic stellar outflow and recover the complex magnetohydrodynamic bow shock flow in the switch-on regime found in De Sterck et al. [Phys. Plasmas, 5, 4015 (1998)]. Copyright © 1999 John Wiley & Sons, Ltd.

Implicit and semi‐implicit schemes: Algorithms

This study formulates general guidelines to extend an explicit code with a great variety of implicit and semi-implicit time integration schemes. The discussion is based on their specific implementation in the Versatile Advection Code, which is a general purpose software package for solving systems of non-linear hyperbolic (and/or parabolic) partial differential equations, using standard high resolution shock capturing schemes. For all combinations of explicit high resolution schemes with implicit and semi-implicit treatments, it is shown how second-order spatial and temporal accuracy for the smooth part of the solutions can be maintained. Strategies to obtain steady state and time accurate solutions implicitly are discussed. The implicit and semi-implicit schemes require the solution of large linear systems containing the Jacobian matrix. The Jacobian matrix itself is calculated numerically to ensure the generality of this implementation. Three options are discussed in terms of applicability, storage requirements and computational efficiency. One option is the easily implemented matrix-free approach, but the Jacobian matrix can also be calculated by using a general grid masking algorithm, or by an efficient implementation for a specific Lax–Friedrich-type total variation diminishing (TVD) spatial discretization. The choice of the linear solver depends on the dimensionality of the problem. In one dimension, a direct block tridiagonal solver can be applied, while in more than one spatial dimension, a conjugate gradient (CG)-type iterative solver is used. For advection-dominated problems, preconditioning is needed to accelerate the convergence of the iterative schemes. The modified block incomplete LU-preconditioner is implemented, which performs very well. Examples from two-dimensional hydrodynamic and magnetohydrodynamic computations are given. They model transonic stellar outflow and recover the complex magnetohydrodynamic bow shock flow in the switch-on regime found in De Sterck et al. [Phys. Plasmas, 5, 4015 (1998)]. Copyright © 1999 John Wiley & Sons, Ltd.

Simulation of magnetization reversal in polycrystalline patterned Co elements

The influence of the polycrystalline microstructure on the switching mechanisms of acicular shaped Co elements was investigated using finite element micromagnetics. The Gilbert equation of motion with a Gilbert damping constant α=1 was solved using a semi-implicit time integration scheme. The elements were 200×40 nm2 and 25 nm thick. The grain size is approximately 8 nm, leading to edge irregularities of the same size. The competitive effects of the shape anisotropy and the random, magnetocrystalline anisotropy lead to a magnetization ripple structure with a wavelength of about 100 nm for zero applied field. With increasing applied field, the magnetization ripple becomes more pronounced. At an applied field of 95 kA/m a vortex, originally formed near sharp edge irregularities, moves into the width of the element. The vortex reaches the opposite edge after about one ns. Then a transverse domain structure of a head to head domain wall forms and the reversed domain expands through the entire particle. The magnitude of magnetostatic interaction field for an array of elements with 50 nm gaps was found to be in the range of 8–20 kA/m. The calculated switching field and the magnetostatic interaction field agree well with in situ magnetization experiments in a transmission electron microscope.

Case Study of Erin Using the FSU Nested Regional Spectral Model

Abstract A case study of Hurricane Erin of the 1995 storm season is presented using the recently developed Florida State University (FSU) Nested Regional Spectral Model. The nested regional spectral model uses a perturbation technique similar to that used in the National Centers for Environmental Prediction and European Centre for Medium-Range Weather Forecasts regional spectral models, but with a number of differences such as the use of a Mercator projection. The perturbations are deviations from the FSU Global Spectral Model (FSUGSM) results and are spectrally represented with π-periodic trigonometric basis functions. The perturbations are relaxed at the boundary to approach the global model results. The perturbation time tendencies are solved using a semi-implicit time integration scheme similar to that used in the FSUGSM. The regional model has the same sigma-coordinate vertical structure and physics as the FSUGSM. Implicit horizontal diffusion and time filtering of the perturbations is included. Erin...

A Finite Element Procedure of a Cyclic Thermoviscoplasticity Model for Solder and Copper Interconnects

A finite element procedure using a semi-implicit time-integration scheme has been developed for a cyclic thermoviscoplastic constitutive model for Pb-Sn solder and OFHC copper, two common metallic constituents in electronic packaging applications. The scheme has been implemented in the commercial finite element (FE) code ABAQUS (1995) via the user-defined material subroutine, UMAT. Several single-element simulations are conducted to compare with previous test results, which include monotonic tensile tests, creep tests, and a two-step ratchetting test for 62Sn36Pb2Ag solder; a nonproportional axial-torsional test and a thermomechanical fatigue (TMF) test for OFHC copper. At the constitutive level, we also provide an adaptive time stepping algorithm, which can be used to improve the overall computation efficiency and accuracy especially in large-scale FE analyses. We also compare the computational efforts of fully backward Euler and the proposed methods. The implementation of the FE procedure provides a guideline to apply user-defined material constitutive relations in FE analyses and to perform more sophisticated thermomechanical simulations. Such work can facilitate enhanced understanding thermomechanical reliability issue of solder and copper interconnects in electronic packaging applications.

André Robert (1929–1993): His Pioneering Contributions to Numerical Modelling

ABSTRACT The pioneering contributions of André Robert to numerical modelling are reviewed. Highlights include: his early work on spectral modelling; the introduction of a weak time filter to permit extended integrations of models for climate modelling; the development of highly-efficient semi-implicit semi-Lagrangian time integration schemes; the extension of these techniques for efficiently integrating the hydrostatic primitive equations to the fully-elastic Euler equations; and the identification of and solution to deficiencies in the traditional approaches to applying lateral boundary conditions in limited area models. These contributions all have one thing in common. They are the result of having reduced a complex problem to its essence in order to properly understand its origin, and thereby develop a conceptually simple solution that not only addresses the problem in its simplified context but also in the original complex one. His inspirational contributions have profoundly influenced, and continue to influence, the development by the international geophysical fluid dynamics community of atmospheric and oceanic models for applications and research in weather, climate and air quality. André Robert was a true pioneer of numerical modelling and Will be sorely missed, but his contributions Will endure for many years to come.

Numerical simulations of mantle convection: Time-dependent, three-dimensional, compressible, spherical shell

Abstract We describe nonlinear time-dependent numerical simulations of whole mantle convection for a Newtonian, infinite Prandtl number, anelastic fluid in a three-dimensional spherical shell for conditions that approximate the Earth's mantle. Each dependent variable is expanded in a series of 4,096 spherical harmonics to resolve its horizontal structure and in 61 Chebyshev polynomials to resolve its radial structure. A semiimplicit time-integration scheme is used with a spectral transform method. In grid space there are 61 unequally-spaced Chebyshev radial levels, 96 Legendre colatitudinal levels, and 192 Fourier longitudinal levels. For this preliminary study we consider four scenarios, all having the same radially-dependent reference state and no internal heating. They differ by their radially-dependent linear viscous and thermal diffusivities and by the specified temperatures on their isothermal, impermeable, stress-free boundaries. We have found that the structure of convection changes dramatically a...

Semi‐implicit finite element methods applied to the solution of the shallow water equations

Semi‐implicit time integration schemes have been combined with finite element space discretization in developing a numerical scheme for the solution of the classical shallow water equations. The principal advantages of this approach are twofold: first, a highly variable gridding capability that allows selective regions of high resolution to be constructed within the computing domain, and second, the use of large time steps chosen on the basis of accuracy rather than by a Courant‐type restriction on their magnitude. The method will be useful in computing the low‐frequency–long wave motions driven by tides and wind on continental shelves and in marginal seas where higher grid resolution is necessary to describe adequately the nearshore dynamics. Application of the methodology is demonstrated by the calculation of the currents and sea surface heights in a hypothetical shelf domain that is driven by a single consituent deep ocean tide.

A Multi-Level Spectral Model. I. Formulation and Hemispheric Integrations

The formulation of a multi-level spectral model suitable for simulation of atmospheric flow on a hemispheric or global scale is presented. The derived primitive equations are employed together with spectral-grid transform procedures in the multi-level domain. An efficient semi-implicit time integration scheme is detailed and results of numerical integrations initialized from analytic fields and Southern Hemisphere data sets are presented. A simple initializing device of divergence dissipation is suggested and shown to be most effective in eliminating spurious large-scale inertia-gravity oscillations.