Explicit Time Integration Scheme Research Articles

Towards optimal use of the explicit [formula omitted]-Bathe time integration method for linear and nonlinear dynamics

In an earlier publication, we proposed a new explicit time integration scheme, the β1/β2-Bathe method, which is simple in its formulation and showed remarkable accuracy in the solution of problems [1]. A particular strength of the method is that it can directly be used as a first-order or second-order scheme by a change of the values of β1 and β2. While good results are obtained with reasonable values of β1 and β2, for excellent accuracy better values of the parameters need to be chosen. We propose in this paper values of β1 and β2 for the first-order scheme, best used in wave propagation analyses, and separate values for the second-order scheme, best used in analyses of structural vibrations. In each case, one set of values of (β1,β2) is given and to possibly improve the results only one of the parameters needs to be changed, that is, β1 for wave propagations and β2 for structural vibrations, making the scheme a one-parameter method. Another strength of the procedure is that physical damping can directly be included in the solution, the effect of which on the stability and accuracy of the solutions we analyze in the paper. The use of the solution scheme in nonlinear analysis is, as we show in the paper, a simple extension from linear analysis. Finally, we give various solutions using the explicit β1/β2-Bathe method in linear and nonlinear analyses to illustrate the performance of the method with the given recommendations for its use.

Numerical modelling of thermal jet assisted rock cutting with double PDC cutters

AbstractPreconditioning of rock for drilling operations is a potential method to facilitate the mechanical breakage and mitigate the tool wear. This paper numerically investigates one such preconditioning technique, namely, the thermal jet assisted rock cutting. For this end, a numerical method for solving the governing thermo‐mechanical problem is developed and validated. The continuum approach is chosen to describe the rock failure, being based on a damage‐viscoplasticity model with the Drucker–Prager yield surface and the modified Rankine surface as the tensile cut‐off. In the damage part, separate scalar damage variables are employed for tension and compression. The cutters are modelled as rigid bodies and their interaction with the rock is modelled by imposing contact constraints by the forward increment Lagrange multiplier method, which is compatible with the chosen explicit time integration scheme. A damage‐based erosion criterion is applied to remove the contact nodes surrounded by heavily damaged elements. The thermal jets are modelled with a moving external heat flux boundary condition. The global thermo‐mechanical problem is solved with a staggered approach explicitly in time while applying mass scaling to increase the critical time step. The novel 3D numerical simulations involving two cutters demonstrate the capabilities of the method with a special emphasis on cutter‐thermal jet configurations. Therefore, the present method provides a potential tool for the bit design in thermal jet assisted drilling.

Simulation of the Dynamic Responses of Layered Polymer Composites under Plate Impact Using the DSGZ Model

This paper presents a numerical study on the dynamic response and impact mitigation capabilities of layered ceramic–polymer–metal (CPM) composites under plate impact loading, focusing on the layer sequence effect. The layered structure, comprising a ceramic for hardness and thermal resistance, a polymer for energy absorption, and a metal for strength and ductility, is analyzed to evaluate its effectiveness in mitigating the impact loading. The simulations employed the VUMAT subroutine of DSGZ material models within Abaqus/Explicit to accurately represent the mechanical behavior of the polymeric materials in the composites. The VUMAT implementation incorporates the explicit time integration scheme and the implicit radial return mapping algorithm. A safe-version Newton–Raphson method is applied for numerically solving the differential equations of the J2 plastic flow theory. Analysis of the simulation results reveals that specific layer configurations significantly influence wave propagation, leading to variations in energy absorption and stress distribution within the material. Notably, certain layer sequences, such as P-C-M and C-P-M, exhibit enhanced impact mitigation with a superior ability to dissipate and redirect the impact energy. This phenomenon is tied to the interactions between the material properties of the ceramic, polymer, and metal, emphasizing the necessity of precise material characterization and enhanced understanding of the layer sequencing effect for optimizing composite designs for impact mitigation. The integration of empirical data with simulation methods provides a comprehensive framework for optimizing composite designs in high-impact scenarios. In the general fields of materials science and impact engineering, the current research offers some guidance for practical applications, underscoring the need for detailed simulations to capture the high-strain-rate dynamic responses of multilayered composites.

Error analysis of second-order local time integration methods for discontinuous Galerkin discretizations of linear wave equations

This paper is dedicated to the full discretization of linear wave equations, where the space discretization is carried out with a discontinuous Galerkin method on spatial meshes which are locally refined or have a large wave speed on only a small part of the mesh. Such small local structures lead to a strong Courant–Friedrichs–Lewy (CFL) condition in explicit time integration schemes causing a severe loss in efficiency. For these problems, various local time-stepping schemes have been proposed in the literature in the last years and have been shown to be very efficient. Here, we construct a quite general class of local time integration methods preserving a perturbed energy and containing local time-stepping and locally implicit methods as special cases. For these two variants we prove stability and optimal convergence rates in space and time. Numerical results confirm the stability behavior and show the proved convergence rates.

A phase field-immersed boundary-lattice Boltzmann coupling method for fluid-structure interaction analysis

Fluid-structure interaction (FSI) is critical in numerous engineering and scientific applications. This paper presents a numerical model for FSI simulations that integrates the lattice Boltzmann method (LBM) for fluid dynamics, the phase field method (PFM) for structural deformation and fracture, and the immersed boundary method (IBM) for fluid-solid coupling. The fluid employs an explicit time integration scheme with a smaller time step than the solid, which uses an implicit time integration scheme, satisfying the CFL condition. During the synchronization of fluid-solid interaction information, a multi-substep strategy is applied. The IBM is implemented using the iterative velocity method for boundary treatment. This approach facilitates the information transfer between domains with different time steps, enhancing the coupling accuracy through iterative computation. The effectiveness of the improved immersed boundary-lattice Boltzmann (IB-LBM) has been verified by the classical cylindrical flow case, which effectively prevents the particle penetration phenomenon and improves the calculation accuracy. The phase field-immersed boundary-lattice Boltzmann method (PF-IB-LBM) is subsequently deployed to simulate the deformation of an aluminum plate under hydrostatic pressure and the aftermath of a dam-break impact on an elastic plate. Verification of the coupling method substantiates its feasibility and efficacy. In addition, the proposed PF-IB-LBM is implemented to simulate the entire process of structural deformation and failure in response to fluid dynamics, which further confirms its capability to predict discontinuous mechanical behaviors such as damage and crack propagation under intricate fluid flows. The results emphasize the flexibility and efficiency of the proposed approach, providing a robust framework for solving more complex scenarios in engineering FSI problems.

Thermo-mechanical coupled peridynamics simulation of concrete failure under fire scenarios

A coupled thermo-mechanical bond-based peridynamical (PD) method is developed to simulate thermal cracking processes in concrete. Based on the bond-based PD heat conduction and motion equations, the PD differential operator (PDDO) is used to express the classical computational fluid dynamics basic equations in a non-local integral form, establishing a PD model for the thermo-mechanical coupling problems. In this model, concrete is considered a heterogeneous material composed of aggregates and mortar, with material parameters assumed to be temperature-dependent. The multi-rate explicit time integration scheme is proposed to overcome the multi-scale time problem in coupled thermo-mechanical systems. The model is applied to simulate the transient heat transfer problem of a homogeneous plate. The results show good agreement with the analytical solution, providing evidence for the correctness and accuracy of the proposed coupled numerical method. The damage behavior of the borehole heated concrete plate is analyzed, followed by the simulation of the damage to the concrete plate under a fire scenario. The numerical results accurately predict the temperature distribution within the heterogeneous concrete and the resulting material damage. This study provides new theoretical basis for the fire resistance design and high-temperature performance improvement of concrete materials and structures.

Fast explicit time integration schemes for parabolic problems in mechanics

AbstractWe present a family of fast explicit time integration schemes of first, second and third order accuracy for parabolic problems in mechanics solved via standard numerical methods that have considerable higher computational efficiency versus existing explicit methods of the same order. The derivation of the new explicit schemes is inspired on the finite increment calculus (FIC) procedure used for obtaining stabilized numerical schemes in fluid and solid mechanics. The new (so‐called) explicit FIC‐Time (EFT) schemes allow considerable larger time steps than the standard first order forward Euler (FE) scheme and the second and third order Adams–Bashforth schemes. The comparison with Runge–Kutta schemes also favors the FIC‐Time schemes in terms of the limit time step size (for second order schemes) and the total number of matrix‐vector multiplications per time step (for second and third order schemes). The new first order explicit schemes have a faster convergence to steady‐state than the FE scheme. The accuracy and efficiency of the new EFT schemes are verified in examples of application to the transient heat conduction equation using the finite element method.

An explicit fourth-order accurate compact method for the Allen-Cahn equation

<abstract><p>In this paper, we propose an explicit spatially fourth-order accurate compact scheme for the Allen-Cahn equation in one-, two-, and three-dimensional spaces. The proposed method is based on the explicit Euler time integration scheme and fourth-order compact finite difference method. The proposed numerical solution algorithm is highly efficient and simple to implement because it is an explicit scheme. There is no need to solve implicitly a system of discrete equations as in the case of implicit numerical schemes. Furthermore, when we consider the temporally accurate numerical solutions, the time step restriction is not severe because the governing equation is a second-order parabolic partial differential equation. Computational tests are conducted to demonstrate the superior performance of the proposed spatially fourth-order accurate compact method for the Allen-Cahn equation.</p></abstract>

Investigation of high strain rate effects on strain-hardening cementitious composites using Voronoi-cell lattice models

Strain-hardening cementitious composites (SHCC) have been reported in several studies to exhibit in several studies to exhibit high toughness even under extreme dynamic loads. To understand such dynamic behavior of SHCC, this study extends a previously developed rate-dependent lattice model by introducing a non-local formulation of the fiber-matrix interface behavior. The Voronoi-cell lattice model, representing the cementitious matrix, employs an explicit time integration scheme and a visco-plastic unit to account for the inertia effect and Stefan effect. Additionally, randomly-distributed short fibers were modeled using an analytical model of shear stress transfer at the fiber-matrix interface. The proposed numerical model successfully simulated the direct tensile behavior and multiple cracking patterns of SHCC, revealing distinct rate-dependent failure characteristics. Through a simple parameter fitting process, the model accurately predicted the dynamic increase factor (DIF) of strength and strain capacity of high-speed tensile experiments. Furthermore, the numerically evaluated rate dependency of the fiber-matrix interface exhibited identical trends to previous experimental studies. Finally, the influences of fluctuations in the applied strain rate during actual high-speed loading tests and the random dispersion of fibers were assessed.

A highly efficient computational approach for fast scan-resolved simulations of metal additive manufacturing processes on the scale of real parts

This article proposes a novel high-performance computing approach for the prediction of the temperature field in powder bed fusion (PBF) additive manufacturing (AM) processes. In contrast to many existing approaches to part-scale simulations, the underlying computational model consistently resolves physical scan tracks without additional heat source scaling, agglomeration strategies or any other heuristic modeling assumptions. A growing, adaptively refined mesh accurately captures all details of the laser beam motion. Critically, the fine spatial resolution required for resolved scan tracks in combination with the high scan velocities underlying these processes mandates the use of comparatively small time steps to resolve the underlying physics. Explicit time integration schemes are well-suited for this setting, while unconditionally stable implicit time integration schemes are employed for the interlayer cool down phase governed by significantly larger time scales. These two schemes are combined and implemented in an efficient fast operator evaluation framework providing significant performance gains and optimization opportunities. The capabilities of the novel framework are demonstrated through realistic AM examples on the centimeter scale including the first scan-resolved simulation of the entire NIST AM Benchmark cantilever specimen, with a computation time of less than one day. Apart from physical insights gained through these simulation examples, also numerical aspects are thoroughly studied on basis of weak and strong parallel scaling tests. As potential applications, the proposed thermal PBF simulation approach can serve as a basis for microstructure and thermo-mechanical predictions on the part-scale, but also to assess the influence of scan pattern and part geometry on melt pool shape and temperature, which are important indicators for well-known process instabilities.

Improved explicit quartic B-spline time integration scheme for dynamic response analysis of viscoelastic systems

A quartic B-spline based explicit scheme is extended for dynamic analysis of viscoelastic systems. In the calculation formulation, different integration approximation schemes for exponential damping models and other damping models are presented. The mathematical derivation of the convolution solving strategy is formulated for viscoelastic systems. For the damping models composed of exponential function, this scheme only requires once convolution calculation in each time step, even though the dynamic equations are used twice. This significantly reduces the calculation time for the damping models composed of exponential function. Numerical examples show that the proposed quartic B-spline based explicit scheme achieves higher accuracy and efficiency compared to other competitive explicit schemes, not only for damping models composed of exponential functions, but also for other viscoelastic damping models.

MPM vs. CEL: Numerical modelling of penetration processes

MPM vs. CEL: Numerical modelling of penetration processes

Estimation of Responses of Spacecraft to High Frequency Shock Loads and Comparison with Test Results

To arrive at shock levels for the equipment of spacecraft, one needs to have a suitable analytical prediction model. One of the prediction techniques is explicit time integration of the governing equation of motion. This methodology is in use for determining the responses due to an impact where the force of impact is known. But they have not been applied to spacecraft where the force is not known, but the accelerations are known. Also, the frequency range of interest covers higher order modes. In this work theoretical estimation of response of spacecraft to high frequency shock excitation is attempted using an explicit time integration scheme. The parameters required for the response estimation are first established. The influence of dynamic relaxation parameter on the estimated responses is discussed. The acceleration responses of two typical spacecraft are measured during their separation from the vehicle adaptor. The responses are theoretically estimated using the parameters established in this work. In both the spacecraft the estimated results are in very good agreement with the measured results with some differences seen at locations far away from the source.

A time-marching procedure based on a sub-step explicit time integration scheme for non-viscous damping systems

A time-marching procedure based on a sub-step explicit time integration scheme for non-viscous damping systems

Virtual element method for phase field modeling of dynamic fracture

In this paper, we propose a new and efficient virtual element scheme for phase field modeling of the dynamic fracture using an explicit time integration scheme. The explicit time integrator divided the whole problem into two parts, namely, mechanical and damage sub-problems. The former is treated as an elastodynamic equation while the latter is treated as a Poisson equation with reaction terms subjected to irreversibility and bounded constraints. To test the performance of the proposed numerical framework, several benchmark problems are validated and the results are in good agreement with the corresponding numerical and experimental study. Moreover, VEM outperforms FEM in view of memory efficiency and choice of element type.

A hybrid implicit-explicit discontinuous Galerkin spectral element time domain (DG-SETD) method for computational elastodynamics

SUMMARY A hybrid implicit-explicit (IMEX) discontinuous Galerkin spectral element time domain (DG-SETD) algorithm is proposed to simulate 3D elastic wave propagation in inhomogeneous media. In this method, the original problem can be divided into a number of well designed subdomains, and the mesh generation of different subdomains is completely independent, thus allowing flexible spatial discretization of complex geometry. The neighboring subdomains are connected by a Riemann transmission condition (RTC), and spectral elements with different interpolation orders can be used in different subdomains to maximize the computational efficiency of multiscale problems to facilitate parallel computing for different subdomains. In particular, the explicit or implicit time iteration scheme can be appropriately selected for a subdomain according to the size of its mesh elements to increase the global time step increment, thus giving higher computational efficiency: For subdomains with coarse meshes, the explicit time integration scheme is used and the time step increment is limited by the Courant−Friedrichs−Lewy (CFL) stability condition; for subdomain with fine structures and thus fine meshes, an implicit time integration scheme is used so that a large time step increment can be used without affecting the stability. In addition, a jump condition of displacement and velocity is introduced to accurately simulating fractures and faults, including lossless and viscous fractures with plane, curve or cross structures. This avoids the volume modeling of the extremely thin fracture structures, and effectively reduces the number of degrees of freedoms (DoFs) in the discretized system without the loss of accuracy. The accuracy, robustness and efficiency of the DG-SETD algorithm are demonstrated by quantitative comparisons of the waveforms with the commercial finite element software COMSOL.

Formulation and Analysis of a Cubic B-Spline-Based Time Integration Procedure for Structural Seismic Response Analysis

In this study, an explicit time integration scheme based on cubic B-spline interpolation strategy and momentum corrector is extended for structural seismic response analysis. The acquisition strategy for the required seismic load is provided. The cubic B-spline-based scheme with moment corrector is generalized for the nonlinear hysteresis system. The analysis shows that the explicit scheme has more desirable stability and accuracy properties than other competitive schemes. The effectiveness of the scheme is demonstrated by the linear and nonlinear numerical examples.

A mathematical theory for mass lumping and its generalization with applications to isogeometric analysis

Explicit time integration schemes coupled with Galerkin discretizations of time-dependent partial differential equations require solving a linear system with the mass matrix at each time step. For applications in structural dynamics, the solution of the linear system is frequently approximated through so-called mass lumping, which consists in replacing the mass matrix by some diagonal approximation. Mass lumping has been widely used in engineering practice for decades already and has a sound mathematical theory supporting it for finite element methods using the classical Lagrange basis. However, the theory for more general basis functions is still missing. Our paper partly addresses this shortcoming. Some special and practically relevant properties of lumped mass matrices are proved and we discuss how these properties naturally extend to banded and Kronecker product matrices whose structure allows to solve linear systems very efficiently. Our theoretical results are applied to isogeometric discretizations but are not restricted to them.

Application of the partial Dirichlet–Neumann contact algorithm to simulate low-velocity impact events on composite structures

Impact simulations for damage resistance analysis are computationally intensive due to contact algorithms and advanced damage models. Both methods, which are the main ingredients in an impact event, require refined meshes at the contact zone to obtain accurate predictions of the contact force and damage onset and propagation through the material. This work presents the application of the partial Dirichlet-Neumann contact algorithm to simulate low-velocity impact problems on composite structures using High-Performance Computing. This algorithm is devised for parallel finite element codes running on supercomputers, and it is extended to explicit time integration schemes to solve impact problems including damage. The proposed framework is validated with a standard test for damage resistance on fiber-reinforced polymer matrix composites. Moreover, the parallel performance of the proposed algorithm has been evaluated in a mesh of 74M of elements running with 2400 processors.

On the numerical properties of high‐order spectral (Euler‐Bernoulli) beam elements

AbstractIn this paper, the numerical properties of a recently developed high‐order Spectral Euler‐Bernoulli Beam Element (SBE) featuring a C1‐continuous approximation of the displacement field are assessed. The C1‐continuous shape functions are based on two main ingredients, which are an Hermitian interpolation scheme and the use of Gauß‐Lobatto‐Legendre (GLL) points. Employing GLL‐points does not only avoid Runge oscillations, but also yields a diagonal mass matrix when exploiting the nodal quadrature technique as a mass lumping scheme. Especially in high‐frequency transient analyses, where often explicit time integration schemes are utilized, having a diagonal mass matrix is an attractive property of the proposed element formulation. This is, however, achieved at the cost of an under‐integration of the mass matrix. Therefore, a special focus of this paper is placed on the evaluation of the numerical properties, such as the conditioning of the element matrices and the attainable rates of convergence (ROCs). To this end, the numerical behavior of the SBEs is comprehensively analyzed by means of selected benchmark examples. In a nutshell, the obtained results demonstrate that the element yields good accuracy in combination with an increased efficiency for structural dynamics exploiting the diagonal structure of the mass matrix.