Efficient Time Integration Research Articles

Dynamic earthquake rupture simulations on nonplanar faults embedded in 3D geometrically complex, heterogeneous elastic solids

Dynamic propagation of shear ruptures on a frictional interface in an elastic solid is a useful idealization of natural earthquakes. The conditions relating discontinuities in particle velocities across fault zones and tractions acting on the fault are often expressed as nonlinear friction laws. The corresponding initial boundary value problems are both numerically and computationally challenging. In addition, seismic waves generated by earthquake ruptures must be propagated for many wavelengths away from the fault. Therefore, reliable and efficient numerical simulations require both provably stable and high order accurate numerical methods.We present a high order accurate finite difference method for: a) enforcing nonlinear friction laws, in a consistent and provably stable manner, suitable for efficient explicit time integration; b) dynamic propagation of earthquake ruptures along nonplanar faults; and c) accurate propagation of seismic waves in heterogeneous media with free surface topography.We solve the first order form of the 3D elastic wave equation on a boundary-conforming curvilinear mesh, in terms of particle velocities and stresses that are collocated in space and time, using summation-by-parts (SBP) finite difference operators in space. Boundary and interface conditions are imposed weakly using penalties. By deriving semi-discrete energy estimates analogous to the continuous energy estimates we prove numerical stability. The finite difference stencils used in this paper are sixth order accurate in the interior and third order accurate close to the boundaries. However, the method is applicable to any spatial operator with a diagonal norm satisfying the SBP property. Time stepping is performed with a 4th order accurate explicit low storage Runge-Kutta scheme, thus yielding a globally fourth order accurate method in both space and time. We show numerical simulations on band limited self-similar fractal faults revealing the complexity of rupture dynamics on rough faults.

Efficient time integration in multiplicative inelasticity

AbstractIn Shutov et al. (Comput Methods Appl Mech Eng 265:213–225, 2013), the numerical time integration of a famous large strain model of Maxwell fluid type has been considered. The underlying model is based on the multiplicative decomposition of the deformation gradient and includes a Neo‐Hookean hyperelasticity relation as well as an incompressible viscous flow rule. Shutov et al. presented a time stepping algorithm for implicit time integration of the inelastic flow rule, which is based on Euler backward time discretisation, prevents error accumulation and is iteration free. In this contribution, the basic idea of the this approach is applied to more general models of multiplicative viscoelasticity. Here, extended hyperelastic relations including general functions of the first principal invariant of deformation tensors are regarded. An efficient time stepping algorithm is derived, where only one scalar equation for one scalar unknown has to be solved within every time step. The approach is applied to a specific viscoelastic model. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Torsional vibration analysis of bars including secondary torsional shear deformation effect by the boundary element method

In this paper a boundary element method (BEM) is developed for the torsional vibration problem of bars of arbitrary doubly symmetric constant cross section, taking into account the nonuniform warping and secondary torsional shear deformation effects (STSDE). The bar is subjected to arbitrarily distributed or concentrated dynamic torsional loading along its length, while its edges are subjected to the most general torsional and warping boundary conditions. Apart from the angle of twist, the primary angle of twist per unit length is considered as an additional 1-D degree of freedom in order to account for the STSDE in the equations of motion of the bar. The warping shear stress distribution and the pertinent secondary torsional rigidity are computed by satisfying local equilibrium considerations under dynamic conditions without adhering to assumptions of Thin Tube Theory (TTT). By employing a distributed mass model system accounting for rotatory and warping inertia, an initial boundary value and two boundary value problems with respect to the variable along the bar time-dependent 1-D kinematical components, to the primary and secondary warping functions, respectively, are formulated. The latter are solved employing a pure BE method, requiring exclusively boundary discretization of the bar׳s cross section. The numerical solution of the aforementioned initial boundary value problem is performed through a BE method leading to a system of differential equations with displacement only unknowns, which is solved using an efficient direct time integration technique. Additionally, for the free vibrations case, a generalized eigenvalue problem is formulated through a similar BE technique. The accuracy and reliability of the results is assessed by FEM solutions employing solid or shell modelling. Both open- and closed-shaped cross section bars are examined and the necessity to include nonuniform torsional and STSD effects in the dynamic analysis of bars is demonstrated.

Hp-Adaptive time integration based on the BDF for viscous flows

This paper presents a procedure based on the Backward Differentiation Formulas of order 1 to 5 to obtain efficient time integration of the incompressible Navier–Stokes equations. The adaptive algorithm performs both stepsize and order selections to control respectively the solution accuracy and the computational efficiency of the time integration process. The stepsize selection (h-adaptivity) is based on a local error estimate and an error controller to guarantee that the numerical solution accuracy is within a user prescribed tolerance. The order selection (p-adaptivity) relies on the idea that low-accuracy solutions can be computed efficiently by low order time integrators while accurate solutions require high order time integrators to keep computational time low. The selection is based on a stability test that detects growing numerical noise and deems a method of order p stable if there is no method of lower order that delivers the same solution accuracy for a larger stepsize. Hence, it guarantees both that (1) the used method of integration operates inside of its stability region and (2) the time integration procedure is computationally efficient. The proposed time integration procedure also features a time-step rejection and quarantine mechanisms, a modified Newton method with a predictor and dense output techniques to compute solution at off-step points.

A Stable and Efficient Direct Time Integration of the Vector Wave Equation in the Finite-Element Time-Domain Method for Dispersive Media

The finite-element time-domain (FETD) solution of the vector wave equation (VWE) is directly extended to doubly dispersive media using the bilinear transform approach. The proposed formulation is quite general and flexible in the sense of material dispersion model in contrast to limitations of existing convolutional-based approaches. In addition, it can be implemented in a very straightforward and efficient manner. A stability analysis is performed that demonstrates the unconditional stability of the original formulation is preserved in the dispersive case. However, in order to avoid the well-known late-time instabilities that arise in this formulation, an alternative formulation is considered and extended to the dispersive case with the same approach. Several numerical examples are simulated to verify the validity and accuracy of the proposed formulations. The late-time stability of the alternative formulation is tested through a numerical example with 20 million time steps. The solution is completely free from late-time growth.

Efficient Time Integration of Maxwell's Equations with Generalized Finite Differences

We consider the computationally efficient time integration of Maxwell's equations using discrete exterior calculus (DEC) as the computational framework. With the theory of DEC, we associate the degrees of freedom of the electric and magnetic fields with primal and dual mesh structures, respectively. We concentrate on mesh constructions that imitate the geometry of the close packing in crystal lattices that is typical of elemental metals and intermetallic compounds. This class of computational grids has not been used previously in electromagnetics. For the simulation of wave propagation driven by time-harmonic source terms, we provide an optimized Hodge operator and a novel time discretization scheme with nonuniform time step size. The numerical experiments show a significant improvement in accuracy and a decrease in computing time compared to simulations with well-known variants of the finite difference time domain method.

A Robust and Efficient Composite Time Integration Algorithm for Nonlinear Structural Dynamic Analysis

This paper presents a new robust and efficient time integration algorithm suitable for various complex nonlinear structural dynamic finite element problems. Based on the idea of composition, the three-point backward difference formula and a generalized central difference formula are combined to constitute the implicit algorithm. Theoretical analysis indicates that the composite algorithm is a single-solver algorithm with satisfactory accuracy, unconditional stability, and second-order convergence rate. Moreover, without any additional parameters, the composite algorithm maintains a symmetric effective stiffness matrix and the computational cost is the same as that of the trapezoidal rule. And more merits of the proposed algorithm are revealed through several representative finite element examples by comparing with analytical solutions or solutions provided by other numerical techniques. Results show that not only the linear stiff problem but also the nonlinear problems involving nonlinearities of geometry, contact, and material can be solved efficiently and successfully by this composite algorithm. Thus the prospect of its implementation in existing finite element codes can be foreseen.

Numerical methods for plasma physics in collisional regimes

We consider the development of accurate and efficient numerical methods for the solution of the Vlasov–Landau equation describing a collisional plasma. The methods combine a Lagrangian approach for the Vlasov solver with a fast spectral method for the solution of the Landau operator. To this goal, new modified spectral methods for the Landau integral which are capable to capture correctly the Maxwellian steady state are introduced. A particular care is devoted to the construction of Implicit–Explicit and Exponential Runge–Kutta methods that permit to achieve high-order and efficient time integration of the collisional step. Several numerical tests are reported which show the high accuracy of the numerical schemes here presented.

Efficient time integration for discontinuous Galerkin approximations of linear wave equations

We consider the combination of discontinuous Galerkin discretizations in space with various time integration methods for linear acoustic, elastic, and electro‐magnetic wave equations. For the discontinuous Galerkin method we derive explicit formulas for the full upwind flux for heterogeneous materials by solving the Riemann problems for the corresponding first‐order systems. In a framework of bounded semigroups we prove convergence of the spatial discretization. For the time integration we discuss advantages and disadvantages of explicit and implicit Runge–Kutta methods compared to polynomial and rational Krylov subspace methods for the approximation of the matrix exponential function. Finally, the efficiency of the different time integrators is illustrated by several examples in 2D and 3D for electro‐magnetic and elastic waves.

Two- and three-phase horizontal slug flow simulations using an interface-capturing compositional approach

Progress on the development of a general framework for the simulation of turbulent, compressible, multi-phase, multi-material flows is described. It is based on interface-capturing and a compositional approach in which each component represents a different phase/fluid. It uses fully-unstructured meshes so that the latest mesh adaptivity methods can be exploited. A control volume-finite element mixed formulation is used to discretise the equations spatially. This employs finite-element pairs in which the velocity has a linear discontinuous variation and the pressure has a quadratic continuous variation. Interface-capturing is performed using a novel high-order accurate compressive advection method. Two-level time stepping is used for efficient time-integration, and a Petrov–Galerkin approach is used as an implicit large-eddy simulation model. Predictions of the numerical method are compared against experimental results for a five-material collapsing water column test case. Results from numerical simulations of two- and three-phase horizontal slug flows using this method are also reported and directions for future work are also outlined.

Defect-based local error estimators for splitting methods, with application to Schrödinger equations, Part III: The nonlinear case

The present work is concerned with the efficient time integration of nonlinear evolution equations by exponential operator splitting methods. Defect-based local error estimators serving as a reliable basis for adaptive stepsize control are constructed and analyzed. In the context of time-dependent nonlinear Schrödinger equations, asymptotical correctness of the local error estimators associated with the first-order Lie–Trotter and second-order Strang splitting methods is proven. Numerical examples confirm the theoretical results and illustrate the performance of adaptive stepsize control.

Efficient time integration methods based on operator splitting and application to the Westervelt equation

Efficient time integration methods based on operator splitting and application to the Westervelt equation

Finite Element Simulation of Diffusion and Precipitation with Application to Internal Oxidation of Ni-Xwt%Cr Alloys at 950°C

For the simulation of internal oxidation phenomena, different numerical approaches are proposed in the literature based on 1D finite differences or on explicit time integration schemes which need small time-steps leading to very long computation times. The aim of this paper is to detail a multi-dimentional finite element approach which is coupled with an efficient implicit time integration algorithm. The thermodynamic activities and the total mass fractions are both used as principal nodal variables. The use of finite elements rather than finite differences greatly facilitates the meshing of 2D and 3D bodies. Its implicit time-integration allows using much larger time-steps without any degradation of the results. An application is proposed for the modeling of internal oxidation of chromia for Ni-Xwt%Cr alloys at 950°C by considering the barrier effect of precipitates.

Dynamic implicit 3D adaptive mesh refinement for non-equilibrium radiation diffusion

The time dependent non-equilibrium radiation diffusion equations are important for solving the transport of energy through radiation in optically thick regimes and find applications in several fields including astrophysics and inertial confinement fusion. The associated initial boundary value problems that are encountered often exhibit a wide range of scales in space and time and are extremely challenging to solve. To efficiently and accurately simulate these systems we describe our research on combining techniques that will also find use more broadly for long term time integration of nonlinear multi-physics systems: implicit time integration for efficient long term time integration of stiff multi-physics systems, local control theory based step size control to minimize the required global number of time steps while controlling accuracy, dynamic 3D adaptive mesh refinement (AMR) to minimize memory and computational costs, Jacobian Free Newton–Krylov methods on AMR grids for efficient nonlinear solution, and optimal multilevel preconditioner components that provide level independent solver convergence.

Discontinuous Spectral Element Methods for Time- and Space-Fractional Advection Equations

We develop spectral element methods for a time- and space-fractional advection equation of the form $\prescript{}{0}{\mathcal{D}}^{\tau}_{t} u(x,t) + \theta\,\, \prescript{}{0}{\mathcal{D}}^{\nu}_{x} u(x,t) = f(x,t)$, of order $\tau\in (0,1]$, $\nu \in (0,1)$, subject to Dirichlet initial/boundary conditions. We present two spectrally accurate and efficient methods for global discretization of both temporal and spatial terms, instead of employing traditional low-order time-integration methods. To this end, we first develop a Petrov--Galerkin in time and discontinuous Galerkin in space (PG-DG) method, where we carry out the time-integration using a single time-domain spectral method (SM), and we perform the spatial discretization using the discontinuous spectral/$hp$ element method (DSEM). This scheme also leads to a more efficient time-integration when the time-derivative is integer-order, i.e., $\tau=1$. We develop the SM-DSEM scheme based on a new spectral theory for fractional Sturm--Liouville problems...

Fast time implicit–explicit discontinuous Galerkin method for the compressible Navier–Stokes equations

An efficient and robust time integration procedure for a high-order discontinuous Galerkin method is introduced for solving the unsteady compressible Navier–Stokes equations. The time discretization is based on an explicit formulation for the convective fluxes and an implicit formulation for the viscous fluxes. The implicit procedure uses a fast iterative algorithm based on a partial uncoupling of variables in neighboring elements introduced in previous works by the authors. In the present work, the method is associated to a Newton–Krylov Jacobian-free method. This association allows to reduce memory requirements and operation counts by avoiding the complete construction of the Jacobian matrix and solving a problem of reduced size. Numerical examples in two space dimensions indicate that the performance of the present method is seen to be significantly improved in terms of CPU time.

An efficient time-integration method for nonlinear dynamic analysis of solids and structures

This paper presents an efficient time-integration method for obtaining reliable solutions to the second-order nonlinear dynamic problems in structural engineering. This method employs both the backward-acceleration differentiation formula and the trapezoidal rule, resulting in a self-starting, single step, second-order accurate algorithm. With the same computational effort as the trapezoidal rule, the proposed method remains stable in large deformation and long time range solutions even when the trapezoidal rule fails. Meanwhile, the proposed method has the following characteristics: (1) it is applicable to linear as well as general nonlinear analyses; (2) it does not involve additional variables (e.g. Lagrange multipliers) and artificial parameters; (3) it is a single-solver algorithm at the discrete time points with symmetric effective stiffness matrix and effective load vectors; and (4) it is easy to implement in an existing computational software. Some numerical results indicate that the proposed method is a powerful tool with some notable features for practical nonlinear dynamic analyses.

Model reduction for efficient time-integration of spatial flexible multibody models

A reduction method is proposed for efficient time-integration of compliant mechanism models that undergo large deflections. Of particular importance for the modelling of this class of mechanisms is the accurate description of geometric non-linearities, as stiffness characteristics can change significantly during deflection. A finite element-based flexible multibody approach is used to describe the compliant mechanism in terms of independent generalized coordinates. The modelling of large deflections requires a sufficient number of finite elements to ensure that deformations remain small in a co-rotational context. Increasing the number of elements, increases, besides the number of degrees of freedom, the largest eigenfrequency in the model. This reduces the allowable step size for explicit time-integrator methods. The proposed reduction method aims to suppress the high frequency vibrational modes which are not important for the desired simulation results, while retaining the geometric non-linearities in the reduced model. For this purpose we add constraint relations between the independent generalized coordinates. These constraint relations can be linear or non-linear. Both cases are investigated in this paper and are implemented as a fixed and an interpolated basis method, respectively. The effectiveness of the two methods is demonstrated by a simulation of a compliant straight guidance in a gravity field that undergoes large deflection. Both methods can yield accurate results with a significant increase in computational efficiency.

Model reduction for efficient time-integration of planar flexible multibody models

A reduction method is proposed for efficient time-integration of (planar) compliant mechanism models that undergo large deflections. Models for this class of mechanisms have to include an accurate description of geometric non-linearities, as stiffness characteristics can change significantly during deflection. A finite element based flexible multibody approach is used to analyse compliant mechanisms in terms of independent coordinates. Geometric transfer functions are applied to express the configuration and deformed state in terms of the independent coordinates. The modelling of large deflections requires using a sufficient number of finite elements to ensure that deformations remain small in a co-rotational context. Increasing the number of elements, increases, besides the number of degrees of freedom, the largest eigenfrequency in the model. This reduces the allowable step size in explicit time-integrator methods.For the proposed reduction method, we aim to suppress the unwanted high frequency modes to increase the allowable step size. This is accomplished by first choosing, where possible, coordinates that remain small during simulation as independent coordinates. Such coordinates are well suited to be reduced using linear projection methods such as modal projection for suppressing the high frequency modes. The configuration and deformed state of the mechanism is subsequently determined with the geometric transfer functions. Consequently, a significant increase in the allowable step size is realised, while retaining the geometric non-linear effects that are contained in the geometric transfer functions.The effectiveness of the method is demonstrated by two planar examples: a compliant straight guidance that undergoes large deflections and a flexible manipulator.

Molecular Dynamic Simulation of Graphene Reinforced Nanocomposites for Evaluating Elastic Constants

The mechanical properties of graphene/cellulose (GC) nanocomposites have been investigated using molecular dynamics (MD) simulations. The influence of graphene concentrations and agglomeration on elastic constants is studied. A polymer consistent force field (pcff) is used in the analysis. An efficient time integration scheme was implied in calculating the force in each time step from the corresponding position and velocity of the atom. The GC nanocomposites system undergoes NVT (constant number of atoms, volume and temperature) and NPT (constant number of atoms, pressure and temperature) ensemble with an applied uniform strain during MD simulation. The stress-strain responses were evaluated for GC unit cell in order to study the effects of graphene concentrations and agglomeration on Young's modulus. The results show significant improvement in Young's modulus for the GC nanocomposites compared to neat cellulose. The MD simulation results show reasonable agreement with the micromechanics based model and available experimental data.