Fast and accurate analyses of spacecraft dynamics using implicit time integration techniques

In this paper the transmission line equation which describes the transient voltage and current distributions of a lightning stroke is employed. The finite element method is used to derive the element equations and one-dimensional linear elements are used to discretize the field region. The implicit Newmark time integration technique is used to convert the resulting second-order ordinary differential equations into a set of recurrence equations which are then solved at each time step. A numerical example is included and discussed.

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 shape design sensitivity analysis (DSA) and optimization of structural transient dynamics are proposed for the finite deformation elastoplastic materials under impact with a rigid surface. A shape variation of the structure is considered using the material derivative approach in continuum mechanics. Hyperelasticitybased multiplicatively decomposed elastoplasticity is used for the constitutive model. The implicit Newmark time integration scheme is used for the structural dynamics. The design sensitivity equation is solved at each converged time step with the same tangent stiffness matrix as response analysis without iteration. The cost of sensitivity computation is more efficient than the cost of response analysis for the implicit time integration method. The efficiency and the accuracy of the proposed method are shown through the design optimization of a vehicle bumper. *Communicated by D. Tortorelli.

This paper presents a new finite element formulation for the solution of transient electromagnetic field problems in multiply-excited magnetic systems. The formulation given here consists of obtaining the magnetic vector potentials and current density distributions at each time instant by applying an implicit time integration technique to the diffusion equation and Ampere's Law, followed by a Galerkin's projection, given the total measurable currents flowing in the system as a function of time. The technique is illustrated by way of solving two diffusion problems. Where possible, computed results are compared with closed form solutions and published data, and excellent agreement between them is found.

We study numerical methods for time-dependent partial differential equations describing wave propagation, primarily applied to problems in quantum dynamics governed by the time-dependent Schrodinger equation (TDSE). We consider both methods for spatial approximation and for time stepping. In most settings, numerical solution of the TDSE is more challenging than solving a hyperbolic wave equation. This is mainly because the dispersion relation of the TDSE makes it very sensitive to dispersion error, and infers a stringent time step restriction for standard explicit time stepping schemes. The TDSE is also often posed in high dimensions, where standard methods are intractable.The sensitivity to dispersion error makes spectral methods advantageous for the TDSE. We use spectral or pseudospectral methods in all except one of the included papers. In Paper III we improve and analyse the accuracy of the Fourier pseudospectral method applied to a problem with limited regularity, and in Paper V we construct a matrix-free spectral method for problems with non-trivial boundary conditions. Due to its stiffness, the TDSE is most often solved using exponential time integration. In this thesis we use exponential operator splitting and Krylov subspace methods. We rigorously prove convergence for force-gradient operator splitting methods in Paper IV. One way of making high-dimensional problems computationally tractable is low-rank approximation. In Paper VI we prove that a splitting method for dynamical low-rank approximation is robust to singular values in the approximation approaching zero, a situation which is difficult to handle since it implies strong curvature of the approximation space.

In order to calculate the time history of the trajectory properties for an optimal continuous-thrust orbit transfer, the equations of motion and adjoint equations are integrated. For optimality, boundary conditions on the Hamiltonian must be satisfied, subject to meeting the initial and final values of the state vectors. At the beginning of the integration, the initial costate variables for the adjoint equations are needed. However, the initial costate variables are usually unknown and thus it is necessary to find these variables. In this paper, evolutionary algorithms (EAs) are introduced and applied to find the values of the initial costate variables. Results obtained from the EAs are compared with the results previously published, and it was found that for several low Earth orbit to geostationary Earth orbit transfers, the initial values of the adjoints are comparable to those found using traditional optimization methods. While many traditional optimization methods are very sensitive to initial values of the adjoint variables, the EAs are able to find the initial values of the adjoints from lower and upper bounds that are up to ±25% away from the actual values.

A coupled implicit-explicit time integration method for compressible unsteady flows

SummaryTo improve the computational efficiency of the numerical manifold method for discontinuous deformation simulations, a spatial‐domain coupled explicit‐implicit time integration algorithm is proposed. A subdomain partition algorithm based on a super manifold element is developed for the numerical manifold method to simulate dynamic motions of blocky rock mass. In different subdomains, explicit or implicit time integration method is employed respectively based on its contact and motion status. These subdomains interact through assembling the corresponding explicit or implicit time integration‐based matrices of different rock blocks. The computational efficiency of the discontinuity system under dynamic loading is improved by partially diagonalizing the global matrices. Two verification examples of a sliding block along an inclined plane under a horizontal acceleration excitation and a multiblock system acted on by dynamic forces are studied to examine the accuracy of the proposed numerical method, respectively. A highly fractured rock mass situated on an inclined slope subjected to seismic excitations is then studied to show the computational efficiency of the developed algorithm. The simulated results are in good agreement with those from the versions using purely implicit or explicit time integration algorithm for the numerical manifold method. The computational efficiency is shown to be higher using the proposed algorithm, which demonstrates its potential for application in dynamic analysis of highly fractured rock masses.

In this brief, numerical techniques for an efficient implementation of the spline transcription (ST) method by using the pseudospectral framework are proposed. It intends to develop a new approach to the ST method, analyses with which can be carried out with nearly the same level of computational burden without resorting to a specific type of collocation points as in the pseudospectral method. Therefore, the piecewise continuous polynomial interpolation is converted into a global spline interpolation, which has the same forms as the Lagrange interpolation and is used to derive the formulas for the partial integration and the differentiation. Using these formulas, the nonlinear optimal control problem can be transcribed in a similar manner as in the pseudospectral method and analyzed with numerical techniques and mathematical theories established well for the pseudospectral method. Also, this brief shows the Lagrange and tension spline interpolation can be used for the proposed methods. Numerical accuracy and efficiency of the proposed method are accessed through its applications to the solution of two nonlinear optimal control problems on uniform nodes. The predicted results of states, controls, and costates are well correlated with the optimal solutions. Finally, this brief recommends the use of advanced grid topologies to enhance both accuracy and convergence by using the flexibility in node selections with the proposed methods.

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.

We investigated composite implicit time integration schemes for structural applications. An unconditionally stable, two-sub-step per time step, the Bathe scheme is analyzed. The transient response analysis of a one-dimensional linear model with step excitation is presented. In the conventional Bathe method, the first sub-step is based on the Newmark scheme with the trapezoidal rule with parameter control. The second sub-step employs a three-point Euler backward method. The modified Bathe method uses a three-point trapezoidal rule with two new Newmark parameters β1 and β2 for the second sub-step. In this contribution, we used both methods with a variable time splitting parameter to analyse numerical errors introduced during the time integration process. The response of a clamped-free rod with excitation force applied at free-end is presented with various time splitting ratios. The results are compared with the conventional Newmark method and suggest that the Bathe scheme with equal time sub-step is suitable for linear structural dynamic applications.

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

Time resolved micromagnetics using a preconditioned time integration method

Conserving energy and momentum in nonlinear dynamics: A simple implicit time integration scheme

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