Time Integration Algorithm Research Articles

Simulation of 24,000 Electron Dynamics: Real-Time Time-Dependent Density Functional Theory (TDDFT) with the Real-Space Multigrids (RMG).

We present the theory, implementation, and benchmarking of a real-time time-dependent density functional theory (RT-TDDFT) module within the RMG code, designed to simulate the electronic response of molecular systems to external perturbations. Our method offers insights into nonequilibrium dynamics and excited states across a diverse range of systems, from small organic molecules to large metallic nanoparticles. Benchmarking results demonstrate excellent agreement with established TDDFT implementations and showcase the superior stability of our time integration algorithm, enabling long-term simulations with minimal energy drift. The scalability and efficiency of RMG on massively parallel architectures allow for simulations of complex systems, such as plasmonic nanoparticles with thousands of atoms. Future extensions, including nuclear and spin dynamics, will broaden the applicability of this RT-TDDFT implementation, providing a powerful toolset for studies of photoactive materials, nanoscale devices, and other systems where real-time electronic dynamics is essential.

Seismic Analysis of Nonlinear Reinforced Concrete Frame Structures

Seismic structural analysis attracts significant interest in Brazil, even though much of the country exhibits low seismicity. Some regions in northwestern Brazil, such as most of the state of Acre and the southwest of the state of Amazonas, present ground motion accelerations that require dynamic analysis for earthquake loading. The construction of some special structures, such as nuclear power plants, requires the consideration of their seismic responses, regardless of their location. The internal forces provided by the Response Spectrum Method, in the frequency domain, are not always suitable for the design of reinforced concrete structures. The behavior of concrete is different under compression and tension. The sum of the modal absolute values through SRSS (Square Root of the Sum of Squares) and CQC (Complete Quadratic Combinations) approaches provides maximum absolute internal forces. However, negative or positive values indicate compression and tension, and reinforcement design is greatly influenced by whether stresses are compressive or tensile. The purpose of this research is to present a methodology for the seismic dynamic analysis of nonlinear framed structures, in the time domain, in which the equilibrium is rigorously examined at the end of each time step. The constitutive models adopted for the cyclic behavior of concrete and steel are discussed. A Timoshenko beam element suitable for this nonlinear analysis is presented. A time integration algorithm, based on Newmark's implicit method, is developed using the element's implicit stiffness matrix that considers the nonlinear behavior of the material. Equivalent nodal forces are calculated from accelerograms of known earthquakes. The adoption of the classic Rayleigh damping model is discussed. The numerical implementation is detailed. Examples are presented and conclusions are drawn.

Heterogeneous Time Integration Algorithm for Structural Dynamics with Localized Hysteretic Nonlinearities

This paper proposes a heterogeneous time integration algorithm to analyze the dynamic response of structures with localized hysteretic nonlinearities. The critical point is to treat the whole structure as a combination of a linear substructure governed by the second-order dynamic equation and a nonlinear substructure controlled by the first-order differential formulation. With this partitioning, tailored numerical integration algorithms with heterogeneous time steps are applied directly to calculate the displacements and hysteretic forces from the linear and the nonlinear substructures, respectively. Subsequently, the dynamic responses of the whole structure are solved by a predictor–corrector procedure, where the hysteretic forces and displacement responses are coupled and exchanged to update the partitioning solutions. Furthermore, the energy balance method is derived to verify the stability of the proposed heterogeneous time integration algorithm. Dynamic responses of structures with friction damper, hysteretic models, and bolted joint models are studied to demonstrate the accuracy and efficiency of the proposed algorithm.

A thermodynamically‐based fractional model combined viscoelastic‐viscoplastic‐ductile damage with application to fiber‐reinforced polymer composites

AbstractA thermodynamically‐based fractional viscoelastic‐viscoplastic‐damage constitutive model combined with continuous damage mechanics (CDM) theory was established, in order to describe the rate‐dependent nonlinear behavior of fiber‐reinforced polymer composites (FRPCs). The fractional Helmholtz free energy consists of four contributions: viscoelastic (VE), viscoplastic (VP), hardening and damage, in which the VE and VP parts are constructed by fractional Zener and Scott‐Blair (SB) element forms respectively. The constitutive equation is obtained through Helmholtz free energy for the fractional Zener model, and plastic flow and hardening evolution law are all derived in the process. The ductile damage, coupled to both VE and VP free energy parts, is introduced through fractional damage energy release rates to model the degradation of material properties. The corresponding strain energy release rate and dissipation contributions are also derived. The fractional implicit time integration algorithms of proposed model are presented. The model is applied to validate tests of FRPCs under various loading conditions. The model validation and comparison are presented by simulating experimental data and existing models in the literature. And the corresponding evolution of dissipated energy is discussed to further valid the characterization ability of the model.Highlights A thermodynamical fractional constitutive model was developed for FRPCs. The Helmholtz free‐energy potential for fractional Zener model is adopted. The physical significance of fractional order parameters is explored. Fractional implicit integration algorithm of proposed model is implemented. The validation and comparison of the model are presented under various loads.

Magneto-viscoelastic rod model for hard-magnetic soft rods under 3D large deformation: Theory and numerical implementation

The main purpose of this work is to develop a three-dimensional (3D) viscoelastic rod model for hard-magnetic soft (HMS) rods under large deformation which are widely used active structures in soft robotics. To do so, the Simo’s viscoelasticity theory has been rationally incorporated into the geometrically exact 3D curved rod model. The proposed model includes the deformation modes of axial tension, shear, bending, and torsion, which is applicable to the HMS rods with arbitrarily initial curved and twisted geometries under 3D large deformation. The viscoelastic constitutive equations of the HMS rod in the present formulation are formulated, which include the general relaxation functions. To obtain the expression for the magnetic load, the rotation-based magnetic free energy density is introduced, and the governing equations of the HMS rod with magnetic load and body force are presented. To obtain the numerical implementation, an implicit time integration algorithm that simply extends the generalized-α method for the rotation group, and the corresponding variational formulation and its linearization of the rod model are derived. To validate the model, five numerical examples, including 2D dynamic buckling, 3D static, and 3D dynamic problem are presented. The dynamic problems include the dynamic snap-through behavior of a bistable HMS arch and damped oscillation of a quarter arc cantilever under 3D deformation. The simulation results show good agreement with the results reported in the literature.

SUNDIALS time integrators for exascale applications with many independent systems of ordinary differential equations

Many complex systems can be accurately modeled as a set of coupled time-dependent partial differential equations (PDEs). However, solving such equations can be prohibitively expensive, easily taxing the world’s largest supercomputers. One pragmatic strategy for attacking such problems is to split the PDEs into components that can more easily be solved in isolation. This operator splitting approach is used ubiquitously across scientific domains, and in many cases leads to a set of ordinary differential equations (ODEs) that need to be solved as part of a larger “outer-loop” time-stepping approach. The SUNDIALS library provides a plethora of robust time integration algorithms for solving ODEs, and the U.S. Department of Energy Exascale Computing Project (ECP) has supported its extension to applications on exascale-capable computing hardware. In this paper, we highlight some SUNDIALS capabilities and its deployment in combustion and cosmology application codes (Pele and Nyx, respectively) where operator splitting gives rise to numerous, small ODE systems that must be solved concurrently.

An improved Arbitrary Lagrangian–Eulerian thermal-fluid model by considering powder deposition effects on melting pool during Direct Energy Deposition processes

Directed Energy Deposition (DED) has emerged notably by offering new possibilities for (re-)manufacturing parts. The multi-phase thermal-fluid models that simulate powder stream deposition are computationally too expensive. The mono-phase Arbitrary Lagrangian–Eulerian (ALE) method are limited in prediction due to exclusion of powder deposition parameters. To address this, we propose an improved 3D mono-phase thermal-fluid model that incorporates powder deposition effects and employs a Moving Thermal-Fluid (MTF) framework to accelerate simulations. Mass, energy, and momentum conservation equations are solved using the finite element method (FEM) and an implicit time integration algorithm, and the ALE method tracks the free surface during deposition. The improved ALE allows considering enthalpy and momentum related to powder deposition through the implementation of new source terms in the energy and momentum conservation equations, leading to more accurate predictions without significantly increasing computing time. Numerical investigations into powder deposition parameters, such as powder distribution, powder enthalpy, and powder momentum, are conducted to identify their impact on melting pool prediction. The in-situ and ex-situ measurements of the melting pool are also performed to check the efficiency of the proposed model. The applications highlight the importance of incorporating powder stream effects and demonstrate the proposed model’s computational efficiency compared to the classical ALE model.

A dual-modified implicit time integration method for three-dimensional impact modelling within the framework of the SBFEM

This paper presents a dual-modified implicit time integration method (DMITIM) for three-dimensional impact modelling within the framework of the scaled boundary finite element method (SBFEM). Controlled by a single parameter, this method effectively dissipates high-frequency responses in impact problems without affecting low-frequency responses, ensuring time integration stability. To address the issue of false oscillations during impacts, additional Lagrange multipliers are introduced to supplement the velocity and acceleration corrections at each time integration step. For nonmatching meshes, by leveraging the mesh flexibility of the SBFEM, a rapid and straightforward remeshing method is utilized to establish a node-to-node contact scheme. The numerical results demonstrate that the DMITIM significantly reduces false oscillations in the velocity and contact force during impact events. It also showcases the flexibility of the SBFEM in handling nonmatching meshes. The combination of the SBFEM with the DMITIM provides enhanced simulation accuracy and stability for impact problems compared to the results from ABAQUS using the HHT-α time integration algorithm.

Nonsmooth dynamic modeling and simulation of spatial mechanisms with frictional translational clearance joints

This paper presents a nonsmooth dynamic approach of modeling and computation for spatial mechanism including spatial frictional translational joints (SFTJs) with small clearances. In this research, the dynamic formulation is derived in the Lie group setting, which leads to a coordinate-free and compact formulation. In the following, the normal contact interaction between the slider and guide is described by the complementarity relations between the constraint force in the normal direction. The tangent contact interaction in the SFTJ is characterized by the Coulomb’s friction law in the type of a set-valued map. Based on the horizontal linear complementarity problem (HLCP), the non-smooth dynamics can be established and calculated by combining the Lemke’s algorithm and the RK-MK time integration algorithm. Finally, A spatial crank-slider mechanism with SFTJs is shown as a numerical case. The simulation results demonstrate the correctness and effectiveness of the proposed method which can capture the nonsmooth dynamical behavior of the system.

A clothing pattern generating framework based on scanned human body model

PurposeAccurate and automatic clothing pattern making is very important in personalized clothing customization and virtual fitting room applications. Clothing pattern generating as well as virtual clothing simulation is an attractive research issue both in clothing industry and computer graphics.Design/methodology/approachPhysics-based method is an effective way to model dynamic process and generate realistic clothing animation. Due to conceptual simplicity and computational speed, mass-spring model is frequently used to simulate deformable and soft objects follow the natural physical rules. We present a physics-based clothing pattern generating framework by using scanned human body model. After giving a scanned human body model, first, we extract feature points, planes and curves on the 3D model by geometric analysis, and then, we construct a remeshed surface which has been formatted to connected quad meshes. Second, for each clothing piece in 3D, we construct a mass-spring model with same topological structures, and conduct a typical time integration algorithm to the mass-spring model. Finally, we get the convergent clothing pieces in 2D of all clothing parts, and we reconnected parts which are adjacent on 3D model to generate the basic clothing pattern.FindingsThe results show that the presented method is a feasible way for clothing pattern generating by use of scanned human body model.Originality/valueThe main contribution of this work is twofold: one is the geometric algorithm to scanned human body model, which is specially conducted for clothing pattern design to extract feature points, planes and curves. This is the crucial base for suit clothing pattern generating. Another is the physics-based pattern generating algorithm which flattens the 3D shape to 2D shape of cloth pattern pieces.

A conservative discontinuous-Galerkin-in-time (DGiT) multirate time integration framework for interface-coupled problems with applications to solid–solid interaction and air–sea models

In this paper we extend the DGiT multirate framework, developed in Connors and Sockwell (2022) for scalar transmission problems, to a solid–solid interaction (SSI) problem involving two coupled elastic solids and a coupled air–sea model with the rotating, thermal shallow water equations. In so doing we aim to demonstrate the broad applicability of the mathematical theory and governing principles established in Connors and Sockwell (2022) to coupled problems characterized by subproblems evolving at different temporal scales. Multirate time integration algorithms employing different time steps, optimized for the dynamics of each subproblem, can significantly improve simulation efficiency for such coupled problems. However, development of multirate algorithms is a highly non-trivial task due to the coupling, which can impact accuracy, stability or other desired properties such as preservation of system invariants. DGiT provides a general template for multirate time integration that can achieve these properties. To elucidate the manner in which DGiT accomplishes this task, we fully detail each step in the application of the framework to the SSI and air–sea coupled problems. Numerical examples illustrate key properties of the resulting multirate schemes for both problems.

Simulating shot peening based on a dislocation density-based model with a novel time integration algorithm

Shot peening has been widely used in processing various components since it can bring in residual compressive stress and effectively refine the grain size of impacted area. To simulate grain refinement induced by shot peening, the dislocation density-based model has recently been introduced, however, the existing time integration algorithm is not stable and usually leads to divergent solutions in iterations. In this paper, a novel time integration algorithm is proposed for the dislocation density-based model. Based upon the algorithm, numerical studies on multi-shot AISI4340 steel are carried out with different coverages, velocities, shot diameters, and peening angles. It is shown that the method converges faster than the two-level iteration method, and the predicted dislocation cell structure sizes after shooting are consistent with experimental results. Besides that, increasing coverage can refine the size of a dislocation cell, which is closely dependent on the shot diameter, impact velocity, and angle. Thus, to achieve the desired grain size or the depth of refinement, it is necessary to take the shot diameter and velocity into account simultaneously.

The subloading surface model in hyperelastic-based plasticity with time integration algorithms in intermediate and current configurations

This paper introduces a novel application of subloading surface concept to effectively model the elastoplastic behavior of materials in the finite strain regime. The subloading surface approach allows for a smoother and more seamless transition from the elastic to the plastic region. Two time integration schemes are presented in this study, one in the intermediate configuration and the other in the current configuration. In the first time integration scheme, co-rotational objective time rates are not required, and the formulation utilizes the Mandel stress. In the second time integration scheme, implemented in Eulerian description, the Kirchhoff stress defined in the current configuration is utilized.The results obtained for the simple shear deformation case demonstrate the absence of oscillations. Additionally, these results show excellent agreement with experimental data and existing findings from the literature. Furthermore, a computational comparison between the time integration schemes using the subloading surface model and a model discussed in the literature reveals that the proposed algorithms in this paper require significantly fewer increments in the given time integrations. This indicates a higher computational efficiency and reduced computational time for the proposed methods. Overall, the predicted responses of all the analyzed cases from the provided time integrations exhibit excellent agreement with each other, confirming the accuracy and effectiveness of the presented model.

Structure preserving and energy dissipative contact approaches for implicit dynamics

AbstractIn this work, several structure preserving and energy dissipative contact approaches are proposed and evaluated. The time integration schemes considered are general with regard to the version of constraint type, but here the emphasis was on mortar contact. The proposed mortar contact approach conserves both linear and angular momentum for mortar contact in a novel way. The proposed time integration scheme can conserve energy or provide strict contact dissipation. In addition, the proposed scheme enforces both gap constraints and gap velocity constraints (i.e., persistency). Using a midstep time integrator often causes energy dissipation during initial impact, here this energy can be recovered in a novel way. Enforcing the gap velocity constraint mitigates the contact chatter of the contact pressure and nodes in many problems. Whereas some approaches enforce the persistency condition and gap constraints simultaneously during the solution of the equations of motion (EOM) requiring multipliers for both constraints included in the equation set, here the gap constraint is solved through the equations of motion and the persistency condition is satisfied in the time integration scheme by the velocity update after the equations of motion. It is shown that this approach is strictly dissipative in that a plastic contact condition can be achieved. Analogous to a coefficient of restitution for rigid bodies, any dissipated energy can then be returned upon release if energy conservation is desired. Structure preserving methods are good for long‐time dynamics simulations and energy conserving and strictly dissipative methods can overcome stability issues associated with standard time integration algorithms such as the Newmark method.

Dynamic modeling of spur gear system considering the coupling effect between bearing deflection and gear mesh stiffness

Bearing deflection has a direct impact on the mesh condition of gear systems via the center distance of the gear pair and corner contact. These factors need to be considered to accurately model the time-varying mesh stiffness and dynamic responses of gears. To address this issue, a spatial coordinate conversion is used to obtain the instantaneous spatial position of the gear pair under the bearing deflection. The tooth pair separation distance is derived through the geometric relationship analysis to model the corner contact effect. A dynamic model is established which employs a step-to-step time integration algorithm that considers the coupling effect between bearing deflection and time-varying mesh stiffness to obtain the dynamic response of the spur gear system. The simulation results of the proposed model show good agreement with the experimental results which demonstrate that the proposed model enables a proper analysis of the influence of bearing deflection on gear mesh behavior. This study provides an original dynamic model for analyzing the dynamic performance of the gear system considering the influence of bearing deflection.

Particle tracking in snow avalanches with in situ calibrated inertial measurement units

Abstract In the course of an artificially triggered avalanche, a particle tracking procedure is combined with supplementary measurements, including Global Navigation Satellite System (GNSS) positioning, terrestrial laser scanning and Doppler radar measurements. Specifically, an intertial measurement unit is mounted inside a rigid sphere, which is placed in the avalanche track. The sphere is entrained by the moving snow, recording translational accelerations, angular velocities and the flux density of Earth's magnetic field. Based on the recorded data, we present a threefold analysis: (i) a qualitative data interpretation, identifying different particle motion phases which are associated with corresponding flow regimes, (ii) a quantitative time integration algorithm, determining the corresponding particle trajectory and associated velocities on the basis of standard sensor calibration, and (iii) an improved quantitative evaluation relying on a novel in situ sensor calibration technique, which is motivated by the limitations of the given dataset. The final results, i.e. the evolution of the angular orientation of the sensor unit, translational and rotational velocities and estimates of the sensor trajectory, are assessed with respect to their reliability and relevance for avalanche dynamics as well as for future design of experiments.

Momentum and near-energy conserving/decaying time integrator for beams with higher-order interpolation on SE(3)

This study aims to develop a simple yet robust time-integration scheme for configuration-interpolated beam finite elements. Geometrically exact theory is employed to model the beams. It utilises the rotations which belong to a non-commutative Lie group and thus require special attention. The configuration is interpolated using a two-node SE(3) interpolation or its generalised implicit variant which enables higher orders. Such interpolation treats position and orientation as a unit and a member of the SE(3) Lie group. This unified approach allows for elegant mathematical manipulation. By adapting the Lie midpoint rule appropriately, it becomes possible to express energy changes in terms of inertial and internal forces, thus enabling the derivation of a momentum conserving and almost energy conserving time integration algorithm. The precision of energy conservation depends solely on the length of the finite elements. With further modification, this algorithm can also become an energy-decaying algorithm. Despite the configuration-dependent nature of the interpolation, the need to deal with its derivatives is avoided, which simplifies the implementation. The method is tested using 3D numerical example with finite rotations which confirms, that the method indeed has the conservation properties and is stable and robust.

A robust and generalized effective DAE framework encompassing different methods, algorithms, and model order reduction for linear and nonlinear second order dynamical systems

For linear and nonlinear dynamical problems, we propose the novel development and advancement of the well-known Generalized Single Step Single Solve (GS4) family of second order time-accurate algorithms encompassing the entire class of LMS methods developed over the past 50 years or so with/without controllable numerical dissipation in conjunction with the Differential Algebraic Equation (DAE) framework and Proper Orthogonal Decomposition (POD). Unlike traditional practices which have severe limitations in the ability to provide generality and flexibility and a wide variety of choices to the analyst without losing order of time accuracy, the proposed implementation uniquely not only allows altogether different numerical time integration algorithms within the GS4 family in each of the different subdomains in a single body, but also additionally allows the selection of different space discretized methods such as the Finite Element Method (FEM) and particle methods, and other spatial methods as well in a single analysis. Moreover, the addition of the POD further provides reduction in computational times. Furthermore, unlike existing state of the art, the present framework readily permits a wide array of implicit–implicit, implicit–explicit, and explicit–explicit couplings and the integration of such a technology is of interest here. Consequently, the present DAE-GS4-POD framework has the flexibility of using different spatial methods and different time integration schemes in different subdomains in a selective manner together with Reduced Order Modeling (ROM) to optimize capturing the local and global features of the representative physics. The ROM also employs an iterative convergence check in acquiring sufficient snapshot data to adequately capture the physics to the prescribed accuracy requirements. Such a novelty, and computational features, are not possible with existing state of art and numerical illustrations validate the claim.

A Moment-Fitted Extended Spectral Cell Method for Structural Health Monitoring Applications

The spectral cell method has been shown as an efficient tool for performing dynamic analyses over complex domains. Its good performance can be attributed to the combination of the spectral element method with mesh-independent geometrical descriptions and the adoption of customized mass lumping procedures for elements intersected by a boundary, which enable it to exploit highly efficient, explicit solvers. In this contribution, we introduce the use of partition-of-unity enrichment functions, so that additional domain features, such as cracks or material interfaces, can be seamlessly added to the modeling process. By virtue of the optimal lumping paradigm, explicit time integration algorithms can be readily applied to the non-enriched portion of a domain, which allows one to maintain fast computing simulations. However, the handling of enriched elements remains an open issue, particularly with respect to stability and accuracy concerns. In addressing this, we propose a novel mass lumping method for enriched spectral elements in the form of a customized moment-fitting procedure and study its accuracy and stability. While the moment-fitting equations are deployed in an effort to minimize the lumping error, stability issues are alleviated by deploying a leap-frog algorithm for the solution of the equations of motion. This approach is numerically benchmarked in the 2D and 3D modeling of damaged aluminium components and validated in comparison with experimental scanning laser Doppler vibrometer data of a composite panel under piezo-electric excitation.

A study of effects of different impact loads on the dynamic and elastoplastic behavior in reservoir rocks at the beginning of hydraulic fracturing

Successful hydraulic fracturing is very important in the development of hydrocarbon-bearing formations. The loading introduced by hydraulic fracturing causes deformation and failure, which are related to the damage accumulation and hydraulic fracture initiation process. This study employs a numerical model that considers the dynamic and elastoplastic behaviors in rocks under the influence of impact loads. The acceleration and wave propagation behaviors are quantified using the model. A time integration algorithm is used to ensure numerical accuracy and stability. The effects of loading rate, loading location, and heterogeneity are quantified. Results show that the elastoplastic and dynamic can effectively capture the wavy mechanical responses in the domain. Strain rate, acceleration, and plasticity can all exhibit oscillatory distribution patterns. Increasing the loading rate can magnify acceleration, strain rate, and the maximum plastic strain, while it reduces the range experiencing these induced changes. Changing the loading types and introducing the heterogeneity consideration both largely alter the mechanical response in the domain, and the waveforms of the mechanical parameters are significantly changed. Failure can occur earlier in layers with more elastic mechanical properties. Exerting 50 MPa loading in 0.01 ms can effectively introduce deformation and failures in the reservoir rock. Doubling the loading rate can effectively improve the ability of creating rock failures, which facilitates the following fracture initiation and propagation processes. This study can be a reference for the understanding of near-well and instantaneous rock mechanical behaviors at the beginning of fracturing.