Explicit Time Integration Method Research Articles

An enhanced transient solver with dynamic p‐adaptation and multirate time integration for electromagnetic and multiphysics simulations

AbstractThis paper presents an enhanced discontinuous Galerkin time‐domain (DGTD) solver for the simulation of both pure electromagnetic (EM) and multiphysics problems. By exploring the local variation of the physics, the polynomial expansion order of the physical quantity of interest is allowed to be adjusted locally and dynamically during the real time of a simulation, leading to a dynamic p‐adaptation technique that can provide a sufficient spatial resolution whenever and wherever needed while keeping the computational cost low. On top of the dynamic adaptation, the constraint of the time step size by an explicit time integration method is alleviated through the application of a multirate time integration (MTI) technique, which permits different time step sizes in elements with different sizes and polynomial orders. Together with the dynamic adaptation and MTI techniques, the enhanced DGTD method is both accurate and efficient in space and time and is employed in the numerical investigations of EM and multiphysics problems to demonstrate its performance and application.

Two novel explicit time integration methods based on displacement-velocity relations for structural dynamics

Two novel explicit time integration methods are proposed based on displacement-velocity relations in this paper for structural dynamics. They avoid the factorization of damping and stiffness matrices, and are truly self-starting due to the exclusion of acceleration vectors. The first method employs the motion equation of expanded form and a linear relation of the displacement and velocity vectors. The recommended parameters, derived from linear analysis, enable the method to possess first-order accuracy mostly and second-order accuracy in the absence of numerical and physical damping. The second method adopts the idea of composite methods, and employs two motion equations of different expanded forms per step. Theoretical analysis indicates that this method can achieve a maximum stability limit of 4, and provides a single-parameter optimal scheme, controlled by the degree of numerical dissipation, for this method. The resulting scheme is second-order accurate with the stability limit ranged from 3.5708 to 4 for the undamped case, and some numerical experiments show that it has better numerical performance compared with some up-to-date explicit methods.

Nonlinear dynamic analysis of hysteretic mechanical systems by combining a novel rate-independent model and an explicit time integration method

This paper presents a computational strategy that combines a novel rate-independent phenomenological model with an explicit time integration method to efficiently perform nonlinear dynamic analyses of non-stiffening hysteretic mechanical systems. The novel rate-independent model, developed by specializing a general class of uniaxial phenomenological models, has an algebraic nature, is based on a set of only three parameters having a clear mechanical significance, and can be easily implemented in a computer program. The adopted explicit structure-dependent time integration method, belonging to the Chang’s family of explicit methods, is unconditionally stable for all non-stiffening hysteretic mechanical systems, has a second-order accuracy, does not suffer from numerical damping, and displays a small relative period error for small time step. Furthermore, it does not require iterative procedures and, consequently, does not suffer from convergence issues. Numerical accuracy and computational efficiency of the proposed procedure are assessed by performing several nonlinear time history analyses on hysteretic mechanical systems and comparing the results with those obtained by employing a conventional strategy based on the celebrated Bouc–Wen model, or its modified version, and the widely used Newmark’s constant average acceleration method.

An accurate two‐stage explicit time integration scheme for structural dynamics and various dynamic problems

SummaryIn this article, a two‐stage explicit time integration method is presented for the numerical analysis of various dynamic problems described by second‐order ordinary differential equations in time. The newly proposed method have negligible damping and period errors in the important low‐frequency range and can introduce moderate numerical damping into the spurious high‐frequency range. The computational effort per step required in the new method is the same as the effort needed in the existing two‐stage explicit methods. The proposed method can provide fourth‐order accurate solutions for velocity independent problems and second‐order accurate solutions for velocity dependent problems, which is not possible in the existing methods. The proposed new method is expected to be an excellent tool for numerical analyses of various dynamic problems including structural dynamics.

A simple explicit single step time integration algorithm for structural dynamics

SummaryIn this article, a new single‐step explicit time integration method is developed based on the Newmark approximations for the analysis of various dynamic problems. The newly proposed method is second‐order accurate and able to control numerical dissipation through the parameters of the Newmark approximations. Explicitness and order of accuracy of the proposed method are not affected in velocity‐dependent problems. Illustrative linear and nonlinear examples are used to verify performances of the proposed method.

Higher-order explicit time integration methods for numerical analyses of structural dynamics

In this article, third- and fourth-order accurate explicit time integration methods are developed for effective analyses of various linear and nonlinear dynamic problems stated by second-order ordinary differential equations in time. Two sets of the new methods are developed by employing the collocation approach in the time domain. To remedy some shortcomings of using the explicit Runge-Kutta methods for second-order ordinary differential equations in time, the new methods are designed to introduce small period and damping errors in the important low-frequency range. For linear cases, the explicitness of the new methods is not affected by the presence of non-diagonal damping matrix. For nonlinear cases, the new methods can handle velocity dependent problems explicitly without decreasing order of accuracy. The new methods do not have any undetermined algorithmic parameters. Improved numerical solutions are obtained when they are applied to various linear and nonlinear problems.

Positivity Preserving Limiters for Time-Implicit Higher Order Accurate Discontinuous Galerkin Discretizations

Currently, nearly all positivity preserving discontinuous Galerkin (DG) discretizations of partial differential equations are coupled with explicit time integration methods. Unfortunately, for many problems this can result in severe time-step restrictions. The techniques used to develop explicit positivity preserving DG discretizations cannot, however, easily be combined with implicit time integration methods. In this paper, we therefore present a new approach. Using Lagrange multipliers, the conditions imposed by the positivity preserving limiters are directly coupled to a DG discretization combined with a diagonally implicit Runge--Kutta time integration method. The positivity preserving DG discretization is then reformulated as a Karush--Kuhn--Tucker (KKT) problem, which is frequently encountered in constrained optimization. Since the limiter is only active in areas where positivity must be enforced, it does not affect the higher order DG discretization elsewhere. The resulting nonsmooth nonlinear algebraic equations have, however, a different structure compared to most constrained optimization problems. We therefore develop an efficient active set semismooth Newton method that is suitable for the KKT formulation of time-implicit positivity preserving DG discretizations. Convergence of this semismooth Newton method is proven using a specially designed quasi-directional derivative of the time-implicit positivity preserving DG discretization. The time-implicit positivity preserving DG discretization is demonstrated for several nonlinear scalar conservation laws, which include the advection, Burgers, Allen--Cahn, Barenblatt, and Buckley--Leverett equations.

Stress-Strain State of Steel Structural Details Under Cyclic Loading

Article shows general principles of elastic plastic continuum mechanics, solid body dynamics and time integration methods. The methods for finite element modeling of steel structure details with high strength bolts and their computation using time domain integration under cyclic loading are considered with due regard to various types of nonlinearities and strain rate effects. Comparison of nonlinear static and nonlinear dynamic dynamics methods that implement explicit and implicit time integration methods of dynamic equilibrium equations is made.

Numerical studies on vibration propagation and damping test V1

Earthquakes and aircraft impacts induce vibrations that propagate throughout the entire building and they need to be considered in designing SSCs (Structures, Systems and Components). Mainly linear calculation methods have been in use in design practice and the codes and standards consider damping ratios only for linear structural analyses. Induced vibrations, especially in damaged concrete structures, have not been studied extensively enough for optimization of structural frameworks and/or qualified systems and components. Experimental data on damping properties of damaged reinforced concrete are needed also for benchmarking analysis programs and methods.
 Recently, within IMPACT project, a new type of test series considering vibration propagation has been carried out at VTT. The test target is a reinforced concrete structure with two parallel walls connected to a floor slab. The front wall is additionally supported by triangular shaped side walls which are connected to the floor slab too. The test structure is supported on elastomeric bearing pads, with back pipes effective mainly in compression and with bars effective in tension. In order to obtain information on vibration propagation in damaged concrete structure at different levels of damage grades the same structure was tested six times. At each time the mass of the deformable stainless steel missile was 50 kg. The hit point located in the middle of the front wall. The impact velocity was about 110 m/s in the first four tests (V1A-D) and about 60 m/s in the remaining two tests (V1E and F). In this paper, numerical results on tests V1A and V1F are compared with the corresponding experimental ones.
 The calculated results, such as accelerations, displacements, their response spectra and strains, are compared with experimental measurements. Five finite element (FE) programs are used in computations: Abaqus, Europlexus, LS-DYNA, SOFiSTiK and an in-house code (IHC).
 Most of the FE-codes in the present study use shell elements. In Abaqus and SOFiSTiK non-linear behaviour of shell section is modelled by dividing the cross section into layers. Reinforcements are also modelled as layers. In Europlexus and IHC, an alternative approach is adopted in which the non-linear behaviour of concrete and reinforcement is homogenized beforehand in the shell thickness direction obtaining relations between stress resultants and generalized strains valid for the shell section. In LS-DYNA, 3D solid elements for modelling concrete and beam elements for modelling reinforcements are used.
 Equations of motion are integrated with explicit central difference time integration method, except in SOFiSTiK implicit integration method is used. Modelling and computations with the mentioned FE-programs are made independently of each other. Computations with LS-DYNA are carried out as blind exercises.
 Consideration of the results from benchmarking point of view is still on-going. However it is evident that analysed results follow reasonable well test results in main design parameter level such as maximum displacements, accelerations and strains. Also frequency spectra are estimated reasonably well.

An improved explicit time integration method for linear and nonlinear structural dynamics

In this article, a new explicit time integration method is developed to analyze linear and nonlinear problems of structural dynamics. Like recently developed explicit time integration methods, the new explicit method can also control the amount of numerical dissipation in the high frequency range. The method is explicit in the presence of the damping matrix, if the mass matrix is diagonal. Due to the unconventional approximations of the displacement vector, the new method does not require evaluation of the initial acceleration vector and other acceleration vectors. Linear and nonlinear problems of structural dynamics can be tackled in a consistent manner, and iterative solution finding procedures are not required. Various illustrative problems are used to investigate improved performance of the new explicit method.

A State Space Formulation for the Evaluation of the Pounding Forces During Earthquake

Abstract In recent years, the pounding effect during earthquake is a subject of high significance for structural engineers. In this paper, a state space formulation of the equation of motion is used in a MATLAB code. The pounding forces are calculated using nonlinear viscoelastic impact element. The numerical study is performed on SDOF structures subjected by 1940 EL-Centro and 1977 Vrancea N-S recording. While most of the studies available in the literature are related to Newmark implicit time integration method, in this study the equations of motion in state space form are direct integrated. The time domain is chosen instead of the complex one in order to catch the nonlinear behavior of the structures. The physical nonlinear behavior of the structures is modeled according to the Force Analogy Method. The coupling of the Force Analogy Method with the state space approach conducts to an explicit time integration method. Consequently, the collision is easily checked and the pounding forces are taken into account into the equation of motion in an easier manner than in an implicit integration method. A comparison with available data in the literature is presented.

Explicit Fiber Beam-Column Elements for Impact Analysis of Structures

The solution of impact problems requires advanced computational techniques to overcome the difficulties associated with large short-duration loads. In this case, the explicit time integration method is typically used, since it provides a stable solution for problems such as the analysis of structures subjected to shock and impact loads. However, most explicit-based finite elements were developed for continuum models such as membrane and solid elements, which renders the problem computationally expensive. On the other hand, the development of fiber-based beam finite elements allows for the simulation of the global structural behavior with very few degrees of freedoms, while accounting for the detailed material nonlinearity along the element length. However, explicit-based fiber beam elements have not been properly formulated, in particular for the case of the emerging force-based beam element. In this paper, two developed fiber plane beam elements that consider an explicit time integration scheme for the solution of the dynamic equation of motion are presented. The first element uses a displacement-based formulation, while the second element uses a force-based formulation. For the latter case, a new algorithm that eliminates the need for iterations at the element level is proposed. The developed elements require the use of a lumped mass matrix and a small time increment to ensure numerical stability. No iterations or convergence checks are required, which renders the problem numerically efficient. The developed explicit fiber beam-column models, particularly the force-based element, represents a simple yet powerful tool for simulating the nonlinear complex effect of impact loads on structures accurately while using very few finite elements. The traditional implicit method of analysis typically fails to provide numerical stable behavior for such short time duration problems. Two correlation studies are presented to highlight the efficiency of the developed elements in modelling impact problems where the strain rate effect is considered in the material models. These examples confirm the accuracy and efficiency of the presented elements.

A finite-element coarse-grid projection method for incompressible flow simulations

Coarse grid projection (CGP) methodology is a novel multigrid method for systems involving decoupled nonlinear evolution equations and linear elliptic Poisson equations. The nonlinear equations are solved on a fine grid and the linear equations are solved on a corresponding coarsened grid. Mapping operators execute data transfer between the grids. The CGP framework is constructed upon spatial and temporal discretization schemes. This framework has been established for finite volume/difference discretizations as well as explicit time integration methods. In this article we present for the first time a version of CGP for finite element discretizations, which uses a semi-implicit time integration scheme. The mapping functions correspond to the finite-element shape functions. With the novel data structure introduced, the mapping computational cost becomes insignificant. We apply CGP to pressure-correction schemes used for the incompressible Navier-Stokes flow computations. This version is validated on standard test cases with realistic boundary conditions using unstructured triangular meshes. We also pioneer investigations of the effects of CGP on the accuracy of the pressure field. It is found that although CGP reduces the pressure field accuracy, it preserves the accuracy of the pressure gradient and thus the velocity field, while achieving speedup factors ranging from approximately 2 to 30. The minimum speedup occurs for velocity Dirichlet boundary conditions, while the maximum speedup occurs for open boundary conditions.

Acceleration of the IMplicit–EXplicit nonhydrostatic unified model of the atmosphere on manycore processors

We present the acceleration of an IMplicit–EXplicit (IMEX) nonhydrostatic atmospheric model on manycore processors such as graphic processing units (GPUs) and Intel’s Many Integrated Core (MIC) architecture. IMEX time integration methods sidestep the constraint imposed by the Courant–Friedrichs–Lewy condition on explicit methods through corrective implicit solves within each time step. In this work, we implement and evaluate the performance of IMEX on manycore processors relative to explicit methods. Using 3D-IMEX at Courant number C = 15, we obtained a speedup of about 4× relative to an explicit time stepping method run with the maximum allowable C = 1. Moreover, the unconditional stability of IMEX with respect to the fast waves means the speedup can increase significantly with the Courant number as long as the accuracy of the resulting solution is acceptable. We show a speedup of 100× at C = 150 using 1D-IMEX to demonstrate this point. Several improvements on the IMEX procedure were necessary in order to outperform our results with explicit methods: (a) reducing the number of degrees of freedom of the IMEX formulation by forming the Schur complement, (b) formulating a horizontally explicit vertically implicit 1D-IMEX scheme that has a lower workload and better scalability than 3D-IMEX, (c) using high-order polynomial preconditioners to reduce the condition number of the resulting system, and (d) using a direct solver for the 1D-IMEX method by performing and storing LU factorizations once to obtain a constant cost for any Courant number. Without all of these improvements, explicit time integration methods turned out to be difficult to beat. We discuss in detail the IMEX infrastructure required for formulating and implementing efficient methods on manycore processors. Several parametric studies are conducted to demonstrate the gain from each of the abovementioned improvements. Finally, we validate our results with standard benchmark problems in numerical weather prediction and evaluate the performance and scalability of the IMEX method using up to 4192 GPUs and 16 Knights Landing processors.

A family of second-order fully explicit time integration schemes

A new family of fully explicit time integration methods is proposed, which has second-order accuracy for structures with and without damping. Using a diagonal mass matrix, the suggested scheme remains fully explicit not only for structures with a non-diagonal damping matrix, but also when the internal force vector is a nonlinear function of velocity. The present algorithm has an acceptable domain of stability, and it is self-starting. This technique introduces effectively numerical dissipation to suppress the high-frequency spurious modes, while at the same time the lower modes are not affected too much. In addition, numerical dispersion error of the scheme is considerably smaller than that of the central difference method. Solving several linear and nonlinear problems highlights the superior performance of the authors’ approach. Findings demonstrate that solution time for the suggested scheme is much less than that of the central difference technique. The related algorithm can be easily implemented into programs, which already contain the central difference method.

Explicit time integration of transient eddy current problems

SummaryFor time integration of transient eddy current problems, commonly implicit time integration methods are used, where in every time, step 1 or several nonlinear systems of equations have to be linearized with the Newton‐Raphson method because of ferromagnetic materials involved. In this paper, a generalized Schur complement is applied to the magnetic vector potential formulation, which converts a differential‐algebraic equation system of index 1 into a system of ordinary differential equations with reduced stiffness. For the time integration of this ordinary differential equations system of equations, the explicit Euler method is applied. The Courant‐Friedrich‐Levy stability criterion of explicit time integration methods may result in small time steps. Applying a pseudoinverse of the discrete curl‐curl operator in nonconducting regions of the problem is required in every time step. For the computation of the pseudoinverse, the preconditioned conjugate gradient method is used. The cascaded subspace extrapolation method is presented to produce suitable start vectors for these preconditioned conjugate gradient iterations. The resulting scheme is validated using the nonlinear TEAM 10 benchmark problem.

Nonlinear Dynamic Analysis of Frame Elements Subjected to Blast Explosions

Abstract A finite element method was recently proposed for performing nonlinear analysis of plane frames subjected to blast loads. This method uses an explicit three-parameter time integration method within a total-load, secant-stiffness analysis framework. Rigorous nonlinear sectional analyses are undertaken, considering the strain rate effects and utilizing nonlinear concrete and reinforcement hysteresis models. Shear effects are included through a 2D implementation of the Disturbed Stress Field Model, which is based on a smeared rotating crack conceptualization. In this study, numerical modeling of 18 previously-tested specimens was undertaken to verify the accuracy, reliability, and practicality of this method for blast load conditions. Some specimens were subjected to multiple blast loads, resulting in 24 simulations in total. Most specimens were subjected to very high peak reflected pressures in the range of 0.35 MPa and exhibited significant damage and nonlinearity. Analysis results (obtained from this method and two other methods from the literature) are compared to the experimental responses in terms of peak displacements, stiffnesses, residual displacements, and crack widths. The three advantages of the proposed method were demonstrated: more accurate modeling of reinforced concrete behaviour, simpler modeling requirements, and shorter analysis times. The proposed method was found to simulate the experimental behaviours of the specimens examined with a high degree of accuracy. The solution algorithm provided unconditional numerical stability, and required much shorter analysis times than continuum finite element methods.

A New Explicit Time Integration Scheme for Nonlinear Dynamic Analysis

An explicit time integration method is presented for the linear and nonlinear dynamic analyses of structures. Using two parameters and employing the Taylor series expansion, a family of second-order accurate methods for the solution of dynamic problems is derived. The proposed scheme includes the central difference method as a special case, while damping is shown to exert no effect on the solution accuracy. The proposed method is featured by the following facts: (i) the relative period error is almost zero for specific values of the parameters; (ii) the numerical dissipation contained can help filter out spurious high-frequency components; and (iii) the crucial lower modes are generally unaffected in the integration. Although the proposed method is conditionally stable, it has an appropriate region of stability, and is self-starting. The numerical tests indicate the improved performance of the proposed technique over the central difference method.

High-order implicit residual smoothing time scheme for direct and large eddy simulations of compressible flows

Restrictions on the maximum allowable time step of explicit time integration methods for direct and large eddy simulations of compressible turbulent flows at high Reynolds numbers can be very severe, because of the extremely small space steps used close to solid walls to capture tiny and elongated boundary layer structures. A way of increasing stability limits is to use implicit time integration schemes. However, the price to pay is a higher computational cost per time step, higher discretization errors and lower parallel scalability. In quest for an implicit time scheme for scale-resolving simulations providing the best possible compromise between these opposite requirements, we develop a Runge–Kutta implicit residual smoothing (IRS) scheme of fourth-order accuracy, based on a bilaplacian operator. The implicit operator involves the inversion of scalar pentadiagonal systems, for which efficient parallel algorithms are available. The proposed method is assessed against two explicit and two implicit time integration techniques in terms of computational cost required to achieve a threshold level of accuracy. Precisely, the proposed time scheme is compared to four-stages and six-stages low-storage Runge–Kutta method, to the second-order IRS and to a second-order backward scheme solved by means of matrix-free quasi-exact Newton subiterations. Numerical results show that the proposed IRS scheme leads to reductions in computational time by a factor 3 to 5 for an accuracy comparable to that of the corresponding explicit Runge–Kutta scheme.

A selective mass scaling method for shear wave propagation analyses in nearly incompressible materials

SummaryThis paper describes a selective mass scaling method which is designed for the analysis of wave propagation problems in nearly incompressible materials. The incompressibility of materials leads to a high value of the compressional wave speed, which makes the time step extremely small in explicit time integration method. The proposed selective mass scaling method selects the eigenfrequencies related to volumetric deformation modes to decrease them, while it keeps the shear eigenmodes unchanged. This makes the time step no longer limited by the compressional wave speed but by the shear wave speed. A significant reduction of CPU time is obtained with a good accuracy for transient problems in small strains on free or largely prestressed media. Copyright © 2016 John Wiley & Sons, Ltd.