Explicit Integration Algorithm Research Articles

Asymmetric yield effect evolving with internal variables during continuous large deformations and its FEM validation

In hexagonal close-packed (HCP) metals and alloys, the asymmetric yield effect that strongly evolves with the internal variables during continuous large deformations is clearly evident and significantly influences their formability. This limits their engineering applications and significant scientific issues regarding the macro performance corresponding to microstructural and texture evolution arise. In this study, a powerful stress-invariant-asymmetric-yield function capable of accurately describing the hydrostatic pressure sensitivity (HPS), tension-compression strength differential effect (SDE), and anisotropy is applied. An analytical interpolation approach based on the hardening effect with the function of the internal variable and the Lode parameter of the stress state is developed to build the continuous evolution between two adjacent yield surfaces. A rate-independent large deformation hypoelasto-plastic constitutive relationship adequately considering the asymmetric yield evolution is built. A plastic potential for the non-associated flow rule is introduced to eliminate the plastic dilatancy. The plastic modulus, corrected with the partial derivative of the continuously interpolated yield function with respect to the internal variable that is capable of capturing the evolution effect, is analytically deduced and straightforwardly expressed. The evolving asymmetric yield effect and its constitutive description are then numerically validated, with insignificant errors and high efficiency, by the explicit integration algorithm of finite element simulations of continuous large deformations of the through-thickness and in-plane compressed Zr. The results indicate that (1) for the different levels of pre-strain, the continuously interpolated asymmetric yield function, together with the proposed constitutive relationship, can precisely describe the non-evolving asymmetric yield of HCP-structured Zr as the traditional ones (Yoon et al., 2014; Plunkett et al., 2007) without considering the evolution effect; (2) for continuous large deformation loading, the continuous yield surfaces built by the continuous interpolation of the stress-invariant-asymmetric-yield function and the proposed constitutive relationship, considering the evolution effect, jointly and clearly corrects the strain hardening rates for all loading routes except the X uniaxial tension without the evolution, and the corrected simulations are in better agreement with the theoretical stress-strain data than uncorrected simulations, in particular for the routes along which the yield surfaces clearly evolve; (3) although the in-plane compressed Zr presents a significantly stronger asymmetric yield effect evolving with the accumulative plastic strain compared to the through-thickness compressed Zr during continuous large deformations, the continuously interpolated yield function and the constitutive relationship considering the evolution effect can still precisely describe the strain hardening processes and stress-strain relationships, and accurately capture the strain-softening effect along the equibiaxial tensile route of the in-plane compressed Zr that cannot be captured by the combinations of the traditional yield functions without considering the HPS, SDE, anisotropy, and their evolution and the general constitutive relationships unsuitable for the large deformation asymmetric yield. It is anticipated that the proposed asymmetric yield evolution and its large deformation constitutive description with the high accuracy and extensive applicability on various asymmetric yield functions, single or multiple internal variables, and complicated loading routes of nonlinear plastic behaviors will be used in precise engineering simulations of HCP metals and alloys and the corresponding scientific studies of microscopic mechanisms.

Stress continuity in DEM-FEM multiscale coupling based on the generalized bridging domain method

Concurrent multiscale method is a spatial and temporal combination of two different scale models for describing the micro/meso and macro mixed behaviors observed in strain localization, failure and phase transformation processes, etc. Most of the existing coupling schemes use the displacement compatibility conditions to glue different scale models, which leads to displacement continuity and stress discontinuity for the obtained multiscale model. To overcome stress discontinuity, this paper presented a multiscale method based on the generalized bridging domain method for coupling the discrete element (DE) and finite element (FE) models. This coupling scheme adopted displacement and stress mixed compatibility conditions. Displacements that were interpolated from FE nodes were prescribed on the artificial boundary of DE model, while stresses at numerical integration points that were extracted from DE contact forces were applied on the material transition zone of FE model (the coupling domain and the artificial boundary of FE model). In addition, this paper proposed an explicit multiple time-steps integration algorithm and adopted Cundall nonviscous damping for quasi-static problems. DE and FE parameters were calibrated by DE simulations of a biaxial compression test and a deposition process. Numerical examples for a 2D cone penetration test (CPT) show that the proposed multiscale method captures both mesoscopic and macroscopic behaviors such as sand soil particle rearrangement, stress concentration near the cone tip, shear dilation, penetration resistance vibration and particle rotation, etc, during the cone penetration process. The proposed multiscale method is versatile for maintaining stress continuity in coupling different scale models.

A coupled time integration algorithm for discontinuous deformation analysis using the numerical manifold method

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.

A numerical method for simulating planar 3D multi-fracture propagation in multi-stage fracturing of horizontal wells

To resolve the issue of design for multi-stage and multi-cluster fracturing in multi-zone reservoirs, a new efficient algorithm for the planar 3D multi-fracture propagation model was proposed. The model considers fluid flow in the wellbore-perforation-fracture system and fluid leak-off into the rock matrix, and uses a 3D boundary integral equation to describe the solid deformation. The solid-fluid coupling equation is solved by an explicit integration algorithm, and the fracture front is determined by the uniform tip asymptotic solutions and shortest path algorithm. The accuracy of the algorithm is verified by the analytical solution of radial fracture, results of the implicit level set algorithm, and results of organic glass fracturing experiment. Compared with the implicit level set algorithm (ILSA), the new algorithm is much higher in computation speed. The numerical case study is conducted based on a horizontal well in shale gas formation of Zhejiang oilfield. The impact of stress heterogeneity among multiple clusters and perforation number distribution on multi-fracture growth and fluid distribution among multiple fractures are analyzed by numerical simulation. The results show that reducing perforation number in each cluster can counteract the effect of stress contrast among perforation clusters. Adjusting perforation number in each cluster can promote uniform flux among clusters, and the perforation number difference should better be 1-2 among clusters. Increasing perforation number in the cluster with high in situ stress is conducive to uniform fluid partitioning. However, uniform fluid partitioning is not equivalent to uniform fracture geometry. The fracture geometry is controlled by the stress interference and horizontal principal stress profile jointly.

Validation of underwater explosion response analysis for airbag inflator using a fluid-structure interaction algorithm

Air gun shock systems are commonly used as alternative explosion energy sources for underwater explosion (UNDEX) shock tests owing to their low cost and environmental impact. The airbag inflator of automotive airbag systems is also very useful to generate extremely rapid underwater gas release in lab-scale tests. To overcome the restrictions on the very small computational time step owing to the very fine fluid mesh around the nozzle hole in the explicit integration algorithm, and also the absence of a commercial solver and software for gas UNDEX of airbag inflator, an idealized airbag inflator and fluid mesh modeling technique was developed using nozzle holes of relatively large size and several small TNT charges instead of gas inside the airbag inflator. The objective of this study is to validate the results of an UNDEX response analysis of one and two idealized airbag inflators by comparison with the results of shock tests in a small water tank. This comparison was performed using the multi-material Arbitrary Lagrangian-Eulerian formulation and fluid-structure interaction algorithm. The number, size, vertical distance from the nozzle outlet, detonation velocity, and lighting times of small TNT charges were determined. Through mesh size convergence tests, the UNDEX response analysis and idealized airbag inflator modeling were validated.

General Formulation of Eliminating Unusual Amplitude Growth for Structure-Dependent Integration Algorithms

An unusual amplitude growth of the steady-state response for structures with high natural frequencies was found in explicit or semi-explicit structure-dependent integration algorithms with unconditional stability, second-order accuracy and no overshoot. This paper proposes a general formulation for eliminating such an unusual amplitude growth, by incorporating the load-dependent term into the displacement recursive formula without changing the numerical properties of the integration algorithm. Compared with the existing formulation using the local truncation error for eliminating the unusual amplitude growth, the proposed formulation has the advantages of less symbolic operations, while naturally including the existing formulation. In addition, it is observed that the coefficients of the load-dependent term are proportional to the limit values of the displacement coefficients in the displacement recursive formula as the natural frequency of the system tends to infinity. Then, the general formulation of the load-dependent term is tested for 15 structure-dependent integration algorithms to verify its correctness. Finally, numerical examples of linear and nonlinear multiple-degrees-of-freedom systems illustrate that the general formulation can remove effectively and conveniently the unusual amplitude growth for dynamic response analyses.

Explicit Integration Algorithm of the Bounding Surface Model Based on Swell–Shrink Rules for Cyclic Behaviors of Clay

Integration algorithm is a key to ensure the better application of the dynamic constitutive model. As for the bounding surface model based on swell–shrink rules in this paper, an explicit integration algorithm based on sub-stepping and error control scheme has be described in detail. For being adaptable to the constitutive model, several modifications have been made. The modified explicit algorithm includes two procedures: the first procedure is the second-order forward modified Euler scheme with sub-stepping and error control; the second procedure is the stress correction. However, due to the shrink rule of the bounding surface, the second procedure is not needed at the unloading phase. Finally, several numerical simulations have been conducted for assessing the performance of the explicit algorithm, which demonstrates that the corresponding algorithm is stable and accurate, and can simulate the mechanical characteristics of the clay pretty well.

Non-iterative explicit integration algorithms based on acceleration time history for nonlinear dynamic systems

A two-step explicit acceleration integration method (EAIM2) and a three-step explicit acceleration integration method (EAIM3), which are entirely explicit time integration algorithms, are proposed based on acceleration time history. The computation efforts and costs can be observably reduced on account of avoiding matrix inversion and iteration processes in nonlinear systems. Four nonlinear systems are employed to analyze the EAIM2, the EAIM3, the HHT-$$\upalpha $$ method, the Newmark explicit method and the generally used Newmark method for comparison purposes. The results show that the highest orders of accuracy of the EAIM2 and the EAIM3 are all of second order. The stability of the proposed methods can remain in a critical state in undamped systems. The puny energy ratio and the periodic energy growth and decay manifest that the proposed methods are endowed with favorable nonlinear stability. The amplitude attenuation of the proposed methods is zero. The proposed methods and the CDM possess the same period elongation. The period error of the proposed methods is smaller than that of the Newmark method in the stability interval. The EAIM2 and the EAIM3 possess the lowest computation efforts at the same accuracy level in the above-mentioned integration methods.

Discussion of Paper ‘Improved Explicit Integration Algorithms for Structural Dynamic Analysis with Unconditional Stability and Controllable Numerical Dissipation’ by Chinmoy Kolay & James M. Ricles, Journal of Earthquake Engineering 2017, http://www.tandfonline.com/loi/ueqe20

ABSTRACT Although it was claimed that the MKR-α method can improve the overshoot and nonlinear stability characteristics of the KR-α method, it seems that it still has a high frequency overshoot in steady-state responses and a weak instability. Three examples are applied to numerically illustrate the two adverse properties. A loading-correction term is introduced into the displacement difference equation to remove the adverse overshoot in high frequency steady-state responses. Besides, it is analytically verified that the MKR-α method has an adverse weak instability. Although the problem of high frequency overshoot in steady-state responses can be overcome, there is no way to eliminate the adverse weak instability for both the KR-α method and MKR-α method. Thus, the applications of the two families of integration methods are very limited. It is demonstrated that a high frequency numerical damping is incapable of mitigating the overshoot caused by a weak instability.

Stability analysis of substructure shake table testing using two families of model-based integration algorithms

In implementing a substructure shake table testing (SSTT), the complete structure is divided into the experimental and the analytical substructures, which are driven by shake table and numerically simulated in a computer, respectively. A number of unconditionally stable and explicit integration algorithms referred to as model-based algorithms with highly computational efficiency have been developed to meet the requirement of real-time in conducting SSTT. This study aims to comprehensively investigate the stability of the SSTT system using two recently proposed model-based integration algorithms with controllable numerical energy dissipation, i.e., Kolay-Ricles-α (KR-α) algorithms and modified KR-α (MKR-α) algorithms. The SSTT system for the 2-degree-of-freedom (2-DOF) structures are firstly formulated. In order to take the contribution of the experimental substructure into consideration, the dynamic condensation is adopted to calculate the integration parameters for the analytical substructure. The discrete transfer function of the 2-DOF-SSTT system are derived. The influences of the mass ratio, the frequency ratio and the time step on the stability of the SSTT system are comprehensively investigated by using the discrete control theory. The results show that for larger values of the mass ratio, the frequency ratio and the time step have negative impact on the stability of the SSTT system. In addition, a subfamily of the MKR-α algorithms along with the dynamic condensation can always ensure the SSTT system stable.

A New Family of Explicit Model-Based Integration Algorithms for Structural Dynamic Analysis

A new family of explicit model-based integration algorithms for solving the equations of motion for linear and nonlinear systems is developed. These algorithms are also known as structure-dependent algorithms because the integration parameters are functions of the complete model of the structural system. A variety of numerical properties of the proposed algorithms, including consistency and local truncation error, stability, numerical dispersion and energy dissipation, overshooting, and frequency response under arbitrary excitation, are investigated using the discrete control theory and amplification matrix for linear elastic systems. In addition, the discrete control theory is applied for assessing the stability of the proposed algorithms for nonlinear structural systems. It is observed that the proposed algorithms exhibit the same numerical characteristics as the well-known Newmark family of integration algorithms. Compared with three existing model-based integration algorithms, i.e. the Chen–Ricles, modified Chen–Ricles, and Gui’s algorithms, the proposed algorithms possess more general and versatile numerical features. As a result, the new family of explicit model-based integration algorithms can be potentially used to solve complicated linear and nonlinear structural dynamics problems.

Procedures to build trust in nonlinear elastoplastic integration algorithm: solution and code verification

In the last decades, with the development of a number of nonlinear elastic–plastic integration algorithms, the correctness or accuracy of the underlying solution becomes the main concern, in both academia and industry. Correctness or accuracy can be estimated (and improved) using verification. Verification is one of the main procedures to build trust in the numerical modeling of any phenomena. A full verification process comprises (a) solution verification (calculation verification) and (b) code verification. Presented in this paper are verification procedures for constitutive, elastic–plastic integration algorithms, as used in computational nonlinear solid mechanics. Both explicit and implicit integration algorithms for elastic–plastic constitutive equations are verified using existing and new developed verification technique. Verification techniques used include prescribed solution forcing and Richardson extrapolation. In addition, grid convergence index is applied to estimate the algorithmic uncertainty during the integration process. Verification of elastic–plastic integration algorithms is applied to a number of material models: from simple von Mises perfectly plastic to sophisticated hyperbolic Drucker–Prager with nonlinear Armstrong–Frederick rotational kinematic hardening. Besides, algorithmic uncertainty is estimated with both associative and non-associative material model. In addition, caveats and pitfalls to consider in the code/solution verification processes are deeply discussed.

Explicit multistep time integration for discontinuous elastic stress wave propagation in heterogeneous solids

SummaryA multistep explicit time integration algorithm is presented for tracking the propagation of discontinuous stress waves in heterogeneous solids whose subdomain‐to‐subdomain critical time step ratios range from tens to thousands. The present multistep algorithm offers efficient and accurate computations for tracking discontinuous waves propagating through such heterogeneous solids. The present algorithm, first, employs the partitioned formulation for representing each subdomain, whose interface compatibility is enforced via the method of the localized Lagrange multipliers. Second, for each subdomain, the governing equations of motion are decomposed into the extensional and shear components so that tracking of waves of different propagation speeds is treated with different critical step sizes to significantly reduce the computational dispersion errors. Stability and accuracy analysis of the present multistep time integration is performed with one‐dimensional heterogeneous bar. Analyses of the present algorithm are also demonstrated as applied to the stress wave propagation in one‐dimensional heterogeneous bar and in heterogeneous plain strain problems.

Response to Maxam and Tamma's discussion (EQE‐18‐0306) to Kolay and Ricles's paper, “Development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation”

SummaryThis paper presents the authors' response to the discussion by Dean J. Maxam and Kumar K. Tamma of the paper titled “Development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation.”

An explicit time integration algorithm for linear and non-linear finite element analyses of dynamic and wave problems

PurposeThe purpose of this paper is to propose a new explicit time integration algorithm for solution to the linear and non-linear finite element equations of structural dynamic and wave propagation problems.Design/methodology/approachThe algorithm is completely explicit so that no linear equation system requires solving, if the mass matrix of the finite element equation is diagonal and whether the damping matrix does or not. The algorithm is a single-step method that has the simple starting and is applicable to the analysis with the variable time step size. The algorithm is second-order accurate and conditionally stable. Its numerical stability, dissipation and dispersion are analyzed for the dynamic single-degree-of-freedom equation. The stability of the multi-degrees-of-freedom non-proportional damping system can be evaluated directly by the stability theory on ordinary differential equation.FindingsThe performance of the proposed algorithm is demonstrated by several numerical examples including the linear single-degree-of-freedom problem, non-linear two-degree-of-freedom problem, wave propagation problem in two-dimensional layer and seismic elastoplastic analysis of high-rise structure.Originality/valueA new single-step second-order accurate explicit time integration algorithm is proposed to solve the linear and non-linear dynamic finite element equations. The algorithm has advantages on the numerical stability and accuracy over the existing modified central difference method and Chung-Lee method though the theory and numerical analyses.

Experimental study of seismic response reduction effects of particle damper using substructure shake table testing method

In order to evaluate the effectiveness of particle dampers (PDs) on seismic response reduction of structures, a series of substructure shake table tests have been conducted. A new family of explicit model-based integration algorithms is used to develop the substructure shake table testing (SSTT) method. To implement the SSTT method, a testing system on the basis of the Quanser shake table II is constructed. The effectiveness and accuracy of the proposed SSTT method and the testing system are verified by comparing the results of complete structure shake table tests and the corresponding substructure shake table tests. Mass ratio, which is defined as the ratio of the damper mass to the structural mass and structural damping ratio are taken as two main parameters in the parametric analyses. Twelve cases of PDs, which are categorized into three patterns, with different materials, sizes, and amounts of particles are selected as the experimental substructures. The substructure shake table test results indicate that the overall seismic response reduction effects of three different types of PDs are very close with same mass ratio. Two seismic response reduction ratios in terms of reducing peak and root-mean-square relative structural displacements are defined to quantify the seismic response reduction effects of the dampers. It can be concluded from the experimental studies that both seismic response reduction ratios increase as the mass ratio increases and decreases as the structural damping ratio increases.

Development of an explicit smoothed particle finite element method for geotechnical applications

By using an explicit time integration scheme and the strain smoothing nodal integration technique, a node-based explicit smoothed particle finite element method (eSPFEM) is presented in this paper. Compared to the original particle finite element method (PFEM), important features of the proposed method include (1) an explicit time integration algorithm to solve the dynamic problem, (2) a strain smoothing technique to avoid volumetric locking effects that are characteristic of low-order elements when the medium is incompressible and (3) the use of a nodal integration technique to bypass the transfer of frequent state variable between Gauss points and node points.

Discussion of “development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation” by Chinmoy Kolay and James M. Ricles

SummaryThe time integration method proposed by Kolay and Ricles, which was claimed to be both explicit and unconditionally stable, is shown to be implicit in the sense of requiring the factorization of an effective stiffness matrix where an explicit method needs no solver. Its original derivation procedure employed discrete control theory concepts, which are in fact, equivalent to conventional recurrence relation concepts aiming to match its spectral properties with those of the three‐parameter optimal/generalized‐α method, thus giving rise to an implicit method within the class of linear multistep methods. It is shown that the resulting method possesses several added computational drawbacks due to its derivation procedure, such as additional effective stiffness inversions and a degraded order of accuracy in general.

SALMON: Scalable Ab-initio Light–Matter simulator for Optics and Nanoscience

SALMON (Scalable Ab-initio Light–Mattersimulator for Optics and Nanoscience, http://salmon-tddft.jp) is a software package for the simulation of electron dynamics and optical properties of molecules, nanostructures, and crystalline solids based on first-principles time-dependent density functional theory. The core part of the software is the real-time, real-space calculation of the electron dynamics induced in molecules and solids by an external electric field solving the time-dependent Kohn–Sham equation. Using a weak instantaneous perturbing field, linear response properties such as polarizabilities and photoabsorptions in isolated systems and dielectric functions in periodic systems are determined. Using an optical laser pulse, the ultrafast electronic response that may be highly nonlinear in the field strength is investigated in time domain. The propagation of the laser pulse in bulk solids and thin films can also be included in the simulation via coupling the electron dynamics in many microscopic unit cells using Maxwell’s equations describing the time evolution of the electromagnetic fields. The code is efficiently parallelized so that it may describe the electron dynamics in large systems including up to a few thousand atoms. The present paper provides an overview of the capabilities of the software package showing several sample calculations. Program summaryProgram Title: SALMON: Scalable Ab-initio Light–Matter simulator for Optics and NanoscienceProgram Files doi:http://dx.doi.org/10.17632/8pm5znxtsb.1Licensing provisions: Apache-2.0Programming language: Fortran 2003Nature of problem: Electron dynamics in molecules, nanostructures, and crystalline solids induced by an external electric field is calculated based on first-principles time-dependent density functional theory. Using a weak impulsive field, linear optical properties such as polarizabilities, photoabsorptions, and dielectric functions are extracted. Using an optical laser pulse, the ultrafast electronic response that may be highly nonlinear with respect to the exciting field strength is described as well. The propagation of the laser pulse in bulk solids and thin films is considered by coupling the electron dynamics in many microscopic unit cells using Maxwell’s equations describing the time evolution of the electromagnetic field.Solution method: Electron dynamics is calculated by solving the time-dependent Kohn–Sham equation in real time and real space. For this, the electronic orbitals are discretized on a uniform Cartesian grid in three dimensions. Norm-conserving pseudopotentials are used to account for the interactions between the valence electrons and the ionic cores. Grid spacings in real space and time, typically 0.02 nm and 1 as respectively, determine the spatial and temporal resolutions of the simulation results. In most calculations, the ground state is first calculated by solving the static Kohn–Sham equation, in order to prepare the initial conditions. The orbitals are evolved in time with an explicit integration algorithm such as a truncated Taylor expansion of the evolution operator, together with a predictor–corrector step when necessary. For the propagation of the laser pulse in a bulk solid, Maxwell’s equations are solved using a finite-difference scheme. By this, the electric field of the laser pulse and the electron dynamics in many microscopic unit cells of the crystalline solid are coupled in a multiscale framework.

Closure to “New Unconditionally Stable Explicit Integration Algorithm for Real-Time Hybrid Testing” by Yu Tang and Menglin Lou

Closure to “New Unconditionally Stable Explicit Integration Algorithm for Real-Time Hybrid Testing” by Yu Tang and Menglin Lou