Studies on composite implicit time integration schemes for transient response analysis of a structural model

We present implicit time integration schemes suitable for modeling free surface Stokes flow dynamics with marker in cell (MIC) based spatial discretization. Our target is for example thermal convection surrounded by deformable surface boundaries to simulate the long term planetary formation process. The numerical system becomes stiff when the dynamical balancing time scale for the increasing/decreasing load by surface deformation is very short compared with the time scale associated with thermal convection. Any explicit time integration scheme will require very small time steps; otherwise, serious numerical oscillation (spurious solutions) will occur. The implicit time integration scheme possesses a wider stability region than the explicit method; therefore, it is suitable for stiff problems. To investigate an efficient solution method for the stiff Stokes flow system, we apply first (backward Euler (BE)) and second order (trapezoidal method (TR) and trapezoidal rule—backward difference formula (TR-BDF2)) accurate implicit methods for the MIC solution scheme. The introduction of implicit time integration schemes results in nonlinear systems of equations. We utilize a Jacobian free Newton Krylov (JFNK) based Newton framework to solve the resulting nonlinear equations. In this work we also investigate two efficient implicit solution strategies to reduce the computational cost when solving stiff nonlinear systems. The two methods differ in how the advective term in the material transport evolution equation is treated. We refer to the method that employs Lagrangian update as “fully implicit” (Imp), whilst the method that employs Eulerian update is referred to as “semi-implicit” (SImp). Using a finite difference (FD) method, we have performed a series of numerical experiments which clarify the accuracy of solutions and trade-off between the computational cost associated with the nonlinear solver and time step size. In comparison with the general explicit Euler method, the second order accurate Imp methods reduce total computational cost successfully through the utilization of a large time step without sacrificing accuracy and stability. Moreover, the proposed SImp method is effective in reducing the computational cost associated with evaluating the nonlinear residual while obtaining a solution similar to the Imp method.

The distinct element method as proposed by Cundall and Strack uses the computationally efficient, explicit, central difference time integration scheme. A limitation of this scheme is that it is only conditionally stable, so small time steps must be used. Some researchers have proposed using an implicit time integration scheme to avoid the stability issues arising from the explicit time integrator typically used in these simulations. However, these schemes are computationally expensive and can require a significant number of iterations to form the stiffness matrix that is compatible with the contact state at the end of each time step. In this paper, a new, simple approach for calculating the critical time increment in explicit discrete element simulations is proposed. Using this approach, it is shown that the critical time increment is a function of the current contact conditions. Considering both two‐ and three‐dimensional scenarios, the proposed refined estimates of the critical time step indicate that the earlier recommendations contained in the literature can be unconservative, in that they often overestimate the actual critical time step. A three‐dimensional simulation of a problem with a known analytical solution illustrates the potential for erroneous results to be obtained from discrete element simulations, if the time‐increment exceeds the critical time step for stable analysis.

An adaptive h‐version finite element procedure is presented for a 2D elastodynamic transient analysis using direct integration. The procedure modifies the spatial discretization, i.e. the mesh, in time so that the spatial discretization error can be controlled in each time instant. We analyse an a priori error estimate for the total energy which shows that an a posteriori error estimate developed by Zienkiewicz and Zhu for linear elastostatics can reasonably be extended to dynamic transient analysis. For the time integration, both explicit and implicit time integration schemes can be coupled with the procedure. However, the study in this paper is confined to the use of the Newmark scheme. Numerical examples are used to study the performance of the error estimate and to illustrate the adaptive procedure. Obtained numerical results indicate that: (i) the extension of the Zienkiewicz–Zhu estimate to dynamic transient analysis is useful; (ii) the presented adaptive procedure can, in an efficient way, provide a finite element solution with a user‐controlled accuracy.

A tendon-driven robot inspired from the bird neck is studied in this paper. The objective is to propose numerical integrations for this robot. The classical explicit and implicit Euler integration schemes can be used on this robot. When the elasticity in the tendons is not considered, these schemes are stable and give similar numerical integration results but the explicit implementation is faster. By considering the dynamics of the robot joints on one hand and the dynamics of the motors on the other hand, we propose two Euler integration schemes, explicit and implicit, that take into account the elasticity in the tendons. These schemes rely on the dynamics of the robot joints on the one hand, and on the dynamics of the motors on the other hand and it links them by the tendon elastic model. We observe that with the tendon elasticity model, the explicit integration becomes unstable, depending on the time step, while the implicit one stays stable. The numerical integration that uses this implicit scheme with tendon elasticity allows one to obtain more precise results, while the model of the system is more realistic. Moreover, the evolution of the tendon elongation during a desired motion can be obtained while they are computed in the integration scheme.

An analysis of implicit time integration schemes for wave propagations

Stability of semidiscrete formulations for parabolic problems at small time steps

This study investigates the full penetration simulation of piles from the ground surface, focusing on frictional contact modeling without mesh distortion. To overcome issues related to mesh distortion and improve solution convergence, the Arbitrary Lagrangian–Eulerian (ALE) adaptive mesh technique was implemented within both explicit and implicit time integration schemes. The numerical model was validated against field experiments conducted at Bothkennar, Scotland, using the Imperial College instrumented displacement pile (ICP) in soft clay, where the soil behavior was effectively represented using the modified Cam-Clay model and the Mohr–Coulomb model. The primary objectives of this study are to evaluate the ALE method performance in handling mesh distortion; analyze the effects of soil–pile interface friction, pile dimensions, and various dilation angles on pile resistance; and compare the effectiveness of explicit and implicit time integration schemes in terms of stability, computational efficiency, and solution accuracy. The ALE method effectively modeled pile penetration in Bothkennar clay, validating the numerical model against field experiments. Comparative analysis revealed the explicit time integration method as more robust and computationally efficient, particularly for complex soil–pile interactions with higher friction coefficients.

The objective of this paper is to comparatively assess the performance of various composite implicit time integration methods. Most popular and widely used scheme—the Bathe, and some recently proposed methods like TTBDF, Wen, and NTTBDF are chosen for the purpose of comparative analysis. All the methods are unconditionally stable and show second-order accuracy. The comparison is done by integrating these schemes on a different class of nonlinear dynamic problems. It is shown that all the considered composite schemes require suitable parametric adjustment to attain the optimal performance except for Bathe as it is a parameter-free method.

A three-dimensional unsteady compressible viscous flow solver has been developed, and applied to a low speed centrifugal compressor configuration. Good agreement with experiments have been obtained for a compressor tested at NASA Lewis Research Center. Comparisons with measured surface pressures, and performance data are given. For this compressor, stall was found to occur in the diffuser region. It is demonstrated that this stall may be eliminated, and stable operation may be restored by the use of bleed valves located on the diffuser walls. INTRODUCTION In many army applications (tanks and rotorcraft) small gas turbine engines are often used. The mass flow rate in these engines is small (below 10 Ibs/sec). The compression is provided by one or two centrifugal compressor stages, sometimes in combination with an axial compressor stage. Centrifugal compressors are well suited for handling such low mass flow rates. They are rugged and compact. However, the geometric complexity of the blades and the twisting blade passages give rise to a host of aerodynamic problems such as unsteadiness, establishment of cross flows, flow separation at the trailing edge of the compressor blade, separation over diffuser guide vanes, and unsteady meandering motion of a 'tip vortex' that originates in the vicinity of the gap between the rotor and the casing. In many instances, this unsteadiness leads to rotating stall and surge. Considerable interest exists in the engine community in understanding and eventually controlling these phenomena, particularly compressor stall and surge. Computational fluid dynamics methods provide an efficient way of studying the complex flow phenomena within centrifugal compressors. Thanks to the availability of post-processing and scientific visualization software, the computed flow fields can be extensively analyzed. It is also convenient to test within a numerical model various control strategies such as bleeding, energizing of incoming flow using jets, excitation /wiggling of guide vanes, etc. Passive mechanisms such as slots, slats and flaps can also be easily studied within the scope of a numerical method. These simulations are thus a useful and economical first step to detailed flow control experiments. Finally, CFD simulations provide a wealth of numerical and flow physics information which are useful in developing reduced order models. For these reasons, a 3-D CFD code capable of analyzing centrifugal compressor configurations has been developed by the present investigators. Rather than solving the flow equations in a rotating coordinate system, the present calculations are carried out in an inertial system, with grid velocity terms used to simulate a rotor. Inflow and outflow boundary conditions appropriate for internal flows in general and centrifugal compressors in particular are used. This paper is organized as follows. The flow solution algorithm is first described. Next, results in the form of velocity fields, surface pressure data and performance map are presented for a NASA low-speed centrifugal compressor. The performance of this compressor for off-design low mass flow ratio conditions is subsequently studied. Methods for improving the compressor performance during such conditions are discussed. NUMERICAL FORMULATION The 3-D unsteady compressible Reynolds averaged Navier-Stokes equations are solved numerically using a time marching scheme. This involves solving the governing equations at each time step by marching in time from an initial flow condition with appropriate boundary conditions. The governing equations are cast in a curvilinear body-fitted coordinate system (£,T|,£) and solved using an Alternating Direction Implicit (ADI) scheme 1. This implicit scheme has been validated for a number of fixed wing and rotary wing configurations both in steady and unsteady flight conditions^'*. The partial derivatives in the governing equations are calculated using finite difference approximations. The Graduate Research Assistant, Student Member AIAA. * Regents' professor, Associate Fellow AIAA. Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. inviscid flux terms are calculated using a third order upwind scheme and the viscous stress terms are calculated using standard central difference representations. The details of the upwind formulation are given by Roe. The time derivative is approximated using a two point first order backward difference. The finite difference representation of the governing Navier-Stokes equations at time level 'n' is components at the inflow boundary were set to zero, assuming that there was no swirl at the inlet. It is possible within the framework of the present boundary conditions to prescribe swirl, non-uniform total pressure and total temperature at the inlet. At every time step, the speed of sound a and the normal component of velocity un were found at the inflow boundary from the relations: AT 2a 2a — u = n — u n = I 8 f R + S S + S , T (1) Interior where Aq* is the change in the flow property vector q during adjacent time levels and At is the time step. The operators 8£, 5r| and 8^ are the standard central difference operators. The viscous terms are evaluated explicitly and the inviscid flux vectors F, G and H are calculated implicitly at time level n+1. A three factor approximate factorization is used to factorize the implicit coefficient matrix operator into block tri-diagonal matrices. The factorized system of equations is solved using the Thomas algorithm. Turbulence effects are modeled using the SpalartAllmaras model. INITIAL AND BOUNDARY CONDITIONS The centrifugal compressor is shown in figure 1. The boundary grids are also shown for a single blade passage. The volume grid is made of three segments: inlet region, the rotor, and the diffuser. The following boundary conditions were applied: Periodicity: In the present study, the flow through only a single blade passage has been studied, assuming blade-toblade periodicity. At the periodic boundaries, the flow properties are computed by averaging the properties on either side of the periodic boundary. Solid walls: At the solid walls, the no slip boundary condition was used. The velocity on the walls of the inlet and the diffuser were set to zero, while the velocity for grid points on the compressor blades and the shaft were set equal to Q x r. The pressure, density and temperature values at the solid surfaces were extrapolated from the interior using relationships such as 9p/3n=dT/9n=3p/3n=0, where n represents the direction normal to the solid surface. Inflow Boundary: At this boundary, the stagnation temperature T0 and the total pressure p0 were assumed known. In the simulation of a compressor drawing the air from the atmosphere, these quantities may be set based on ambient conditions. In a multistage compressor analysis, these quantities will be known at the downstream boundary of the previous stage. The tangential (vand w-) (2)

Finite element transient analysis of composite and sandwich plates based on a refined theory and implicit time integration schemes

The present thesis describes the development of a computational method for the numerical simulation of compressible flows in aerodynamics. The main goal is to study high-order discretization schemes that can potentially deliver a better computational efficiency. The spatial part of the conservation equations is discretized with the finite difference method. In particular, Summation By Parts (SBP) finite difference operators and treatment of boundary conditions via Simultaneous Approximation Terms (SAT) have been investigated. An SBP finite difference operator is essentially a (high-order) centered finite difference scheme with a specific closure at the boundary. The SAT are penalty-like terms that enforce the boundary conditions weakly and are used to augment the SBP schemes. The combination of SBP operators with SAT boundary terms constitutes schemes that, for the linearized equations governing a smooth flow, are provably stable (thus convergent) schemes. The discretization in time is carried out with high-order schemes as well. Both implicit and explicit time integration schemes have been investigated. The system of non-linear equations that arises when using the implicit time integration schemes is solved via (a damped) Newton's method. On the target grid, the initial guess required for Newton's method is obtained by performing grid sequencing combined a series of explicit relaxation iterations on each grid level. The flow equations and the equation of the turbulence model (if present) are solved in a fully-coupled manner. Numerical simulations for inviscid flow, laminar flow, as well as simulations based on DNS, LES and RANS have been performed. Results show that the described computational method is particularly well-suited for smooth external flows about relatively simple geometries, solved with the DNS and LES approach. For these types of flows the gain in efficiency obtained by using high-order schemes is clear (more than a factor of 5). For RANS, however, the comparison with a state-of-the-art second order finite volume code suggests that solving the turbulence and the flow equations fully coupled with implicit schemes may not be the most efficient iteration strategy. On the other hand, the numerical simulations of non-smooth/internal flows highlighted the need of further development of the SBP-SAT framework.

Implicit–explicit time integration of a high-order particle-in-cell method with hyperbolic divergence cleaning

The role of time integration in energy conservation in Smoothed Particle Hydrodynamics fluid dynamics simulations

We present a novel particle‐based method for stable simulation of elasto‐plastic materials. The main contribution of our method is an implicit numerical integrator, using a physically‐based model, for computing particles that undergo both elastic and plastic deformations. The main advantage of our implicit integrator is that it allows the use of large time steps while still preserving stable and physically plausible simulation results. As a key component of our algorithm, at each time step we compute the particle positions and velocities based on a sparse linear system, which we solve efficiently on the graphics hardware. Compared to existing techniques, our method allows for a much wider range of stiffness and plasticity settings. In addition, our method can significantly reduce the computation cost for certain range of material types. We demonstrate fast and stable simulations for a variety of elasto‐plastic materials, ranging from highly stiffelastic materials to highly plastic ones.

Time resolved micromagnetics using a preconditioned time integration method