Size Of Integration Time Step Research Articles

A data-driven reduced-order model for stiff chemical kinetics using dynamics-informed training

A data-based reduced-order model (ROM) is developed to accelerate the time integration of stiff chemically reacting systems by effectively removing the stiffness arising from a wide spectrum of chemical time scales. Specifically, the objective of this work is to develop a ROM that acts as a non-stiff surrogate model for the time evolution of the thermochemical state vector (temperature and species mass fractions) during an otherwise highly stiff and nonlinear ignition process. The model follows an encode-forecast-decode strategy that combines a nonlinear autoencoder (AE) for dimensionality reduction (encode and decode steps) with a neural ordinary differential equation (NODE) for modeling the dynamical system in the AE-provided latent space (forecasting step). By means of detailed timescale analysis by leveraging the dynamical system Jacobians, this work shows how data-based projection operators provided by autoencoders can inherently construct the latent spaces by removing unnecessary fast timescales, even more effectively than physics-based counterparts based on an eigenvalue analysis. A key finding is that the most significant degree of stiffness reduction is achieved through an end-to-end training strategy, where both AE and neural ODE parameters are optimized simultaneously, allowing the discovered latent space to be dynamics-informed. In addition to end-to-end training, this work highlights the vital contribution of AE nonlinearity in the stiffness reduction task. For the prediction of homogeneous ignition phenomena for H2-air and C2H4-air mixtures, the proposed ROM achieves several orders-of-magnitude increase in the integration time step size when compared to (a) a baseline CVODE solver for the full-chemical system, (b) statistical technique – principal component analysis (PCA), and (c) computational singular perturbation (CSP), a vetted physics-based stiffness-reducing modeling framework.

Nonstandard Numerical Scheme for a Predator-Prey Model Involving Predator Cannibalism and Refuge

In this study, we implement a Nonstandard Finite Difference (NSFD) scheme for a predator-prey model involving cannibalism and refuge in predator. The scheme which is considered as a discrete dynamical system is analyzed. The performed analysis includes the determination of equilibrium point and its local stability. The system has four equilibrium points, namely the origin, the prey extinction point, the predator extinction point, and the coexistence point, which have exactly the same form and existence conditions as those in continuous system. The local stability of each first three equilibrium points is consistent with the one in continuous system. The stability of the coexistence point depends on the integration time step size. Nevertheless, the NSFD scheme allows us to choose the integration time step size for the solution to converge to a feasible point more flexible than the Euler and 4th order Runge-Kutta schemes. These are shown via numerical simulations.

An implicit asynchronous variational integrator for flexible multibody dynamics

To accurately capture the dynamic responses of flexible multibody systems with complex geometric shapes or with contact and collision problems, extremely fine meshes have to be used. However, most previous algorithms for simulating these systems adopt a single integration time step size. A way to improve computation efficiency is to use different integration time steps for different elements with different sizes. In this study, a novel implicit AVI (asynchronous variational integrator) is originally proposed to study dynamics of complex flexible multibody systems. The proposed integrator allows different integration time step sizes for different elements with different mesh sizes. The multipoint constraint method is utilized to glue non-matching elements. To establish the system’s discrete Euler–Lagrange equations (DELs) , the diagonal-scaling method is used to construct the lumped mass matrix of elements. Finally, seven numerical examples are presented to comprehensively validate accuracy and efficiency of the proposed integrator. Numerical results indicate that the proposed AVI is capable of accurately capturing the dynamic response of flexible multibody systems and preserving systems’ energy and momentum. Compared with the generalized-α algorithm with a single time step size, the proposed AVI can significantly improve the computation efficiency for the dynamic simulation of complex flexible multibody systems with clearance joints.

Extraction of material properties through multi-fidelity deep learning from molecular dynamics simulation

Simulation of reasonable timescales for any long physical process using molecular dynamics (MD) is a major challenge in computational physics. In this study, we have implemented an approach based on multi-fidelity physics informed neural network (MPINN) to achieve long-range MD simulation results over a large sample space with significantly less computational cost. The fidelity of our present multi-fidelity study is based on the integration timestep size of MD simulations. While MD simulations with larger timestep produce results with lower level of accuracy, it can provide enough computationally cheap training data for the MPINN to learn an accurate relationship between these low-fidelity results and high-fidelity MD results obtained using smaller simulation timestep. We have performed two benchmark studies, involving one and two component Lennard-Jones systems, to determine the optimum percentage of high-fidelity training data required to achieve accurate results with high computational saving. The results show that important system properties such as system energy per atom, system pressure and diffusion coefficients can be determined with high accuracy while saving 68% computational costs. Finally, as a demonstration of the applicability of our present methodology in practical MD studies, we have studied the viscosity of argon-copper nanofluid and its variation with temperature and volume fraction by MD simulation using MPINN. Then we have compared them with numerous previous studies and theoretical models. Our results indicate that MPINN can predict accurate nanofluid viscosity at a wide range of sample space with significantly small number of MD simulations. Our present methodology is the first implementation of MPINN in conjunction with MD simulation for predicting nanoscale properties. This can pave pathways to investigate more complex engineering problems that demand long-range MD simulations.

Shape sensitivity analysis for non-differentiable performance measures in multiscale crack propagation simulation using regression hybrid method

ABSTRACTIn this paper, we present a regression hybrid method that calculates shape sensitivity coefficients for multiscale crack propagation problems with performance measures that are non-differentiable in numerical implementation. These measures are crack propagation speed (or crack speed) defined at atomistic level obtained by solving coupled atomistic/continuum structures using the bridging scale method (BSM). The major contributions of this paper are: first, by analyzing the characteristics of the performance measures of crack speed in design space, this paper verifies for the first time that these measures are theoretically continuous and differentiable with respect to design variables, and as a result, the sensitivity coefficients exist in theory; second, to overcome the non-differentiability of the performance measures in numerical computation due to the finite size of integration time step, this paper proposes a regression hybrid method that calculates the shape sensitivity coefficients of crack speed through polynomial regression analysis based on the sensitivity of atomic responses, which is calculated through analytical shape design sensitivity analysis (DSA). And finally, the proposed method supports for 3D crack propagation problems with periodic boundary condition in one direction. A nano-beam example is used to demonstrate numerically the feasibility, accuracy, and efficiency of the proposed method.

Anisotropy in the crystal growth of hexagonal ice, Ih

Growth of ice crystals has attracted attention because ice and water are ubiquitous in the environment and play critical roles in natural processes. Hexagonal ice, I(h), is the most common form of ice among 15 known crystalline phases of ice. In this work we report the results of an extensive and systematic molecular dynamics study of the temperature dependence of the crystal growth on the three primary crystal faces of hexagonal ice, the basal {0001} face, the prism {1010} face, and the secondary prism {1120} face, utilizing the TIP4P-2005 water model. New insights into the nature of its anisotropic growth are uncovered. It is demonstrated that the ice growth is indeed anisotropic; the growth and melting of the basal face are the slowest of the three faces, its maximum growth rates being 31% and 43% slower, respectively, than those of the prism and the secondary prism faces. It is also shown that application of periodic boundary conditions can lead to varying size effect for different orientations of an ice crystal caused by the anisotropic physical properties of the crystal, and results in measurably different thermodynamic melting temperatures in three systems of similar, yet moderate, size. Evidence obtained here provides the grounds on which to clarify the current understanding of ice growth on the secondary prism face of ice. We also revisit the effect of the integration time step on the crystal growth of ice in a more thorough and systematic way. Careful evaluation demonstrates that increasing the integration time step size measurably affects the free energy of the bulk phases and shifts the temperature dependence of the growth rate curve to lower temperatures by approximately 1 K when the step is changed from 1 fs to 2 fs, and by 3 K when 3 fs steps are used. A thorough investigation of the numerical aspects of the simulations exposes important consequences of the simulation parameter choices upon the delicate dynamic balance that is involved in ice crystal growth.

Sensitivity study of gamma‐function pseudodynamic algorithm

The γ‐function explicit method has been developed for pseudodynamic testing since it can have favorable numerical dissipation. This numerical dissipation originates from the use of the structure‐dependent coefficient matrices in a quadrature equation. These coefficient matrices highly depend upon initial stiffness matrix, which is experimentally determined before the pseudodynamic test. Since the initial stiffness matrix determined from an experimental procedure may be different if a different displacement is imposed upon a specified degree of freedom to obtain its corresponding column, the coefficient matrices will vary accordingly. Sensitivity studies of this variation of the coefficient matrices for numerical properties and error propagation properties are conducted herein. It is revealed by both the analytical and numerical results that the performance of the γ‐function explicit pseudodynamic algorithm for both linear elastic and nonlinear systems is insensitive to the variation of the coefficient matrices in the quadrature equation, especially for a small value of the product of the natural frequency and the size of integration time step.

Approach for Overcoming Numerical Inaccuracy Caused by Load Discontinuity

It was found that the discontinuity at the end of an impulse will lead to numerical inaccuracy as this discontinuity will result in an extra impulse and thus an extra displacement in the time history analysis. In addition, this extra impulse is proportional to the discontinuity value at the end of the impulse and the size of integration time step. To overcome this difficulty, an effective approach is proposed to reduce the extra impulse and hence the extra displacement. In fact, the novel approach proposed in this paper is to perform a single small time step immediately upon the termination of applied impulse, whereas other time steps can be conducted by using the step size determined from accuracy consideration in period. The feasibility of this approach is analytically explored. Further, analytical results are confirmed by numerical examples. Numerical studies also show that this approach can be applied to other step-by-step integration methods. It seems that to slightly complicate the programming of dynamic analysis codes is the only disadvantage of this approach.

Application of TRAC/BF1-ENTRÉE-NASCA to OECD NEA/NSC BWR Turbine Trip Benchmark

A coupled plant simulation system TRAC/BF1-ENTRÉE was applied to the Nuclear Energy Agency/National Security Council boiling water reactor turbine trip benchmark. Through regular exercise 3 and extreme scenarios 3 and 4, its adequacy and robustness were validated. It was deduced that the cross-section format and the core boundary conditions are major influential factors causing errors in three-dimensional power predictions. Power swings observed in extreme scenarios were attributed to intermittent void generation and void sweeping driven by rapid pressurization. Based on a series of sensitivity studies for extreme scenario 4, it was confirmed that neglect of in-channel direct heating causes a large positive reactivity insertion and neglect of bypass direct heating causes only a small change in reactivity effects. Specifying an integration time-step size of <1 ms is recommended for keeping the numerical error within an acceptable level. To investigate the detailed in-channel void distribution and its possible influences on the fuel thermal margin, a one-way coupled system between TRAC/BF1-ENTRÉE and the three-field subchannel code NASCA was developed. Detailed void distributions at the upper part of the core where the boiling transition will occur become sufficiently uniform during the major period of the turbine trip event. Their influences on the thermal margin seem negligible.

Macromolecular conformational dynamics in torsional angle space

A Brownian dynamics treatment in torsional angle space is presented for the simulation of conformational dynamics of macromolecules with fixed bond lengths and bond angles and with an arbitrary intramolecular potential energy function. The advantages of the torsional angle space treatment over similar treatments (Brownian dynamics or molecular dynamics) in atomic coordinate space are that, first, the number of variables is reduced by roughly a factor of 10 and, second, the integration time step size is increased by 3 to 4 orders of magnitude (because, by confining the treatment to the torsional angle space, the time step size is not limited by the fast oscillation modes of covalent bonds but rather by the slow motion of macromolecular segments whose time scale is roughly 3 to 4 orders of magnitude larger than that of bond oscillations). Consequently, the exploration of global conformational relaxation processes becomes computationally possible. The treatment is tested by studying the folding kinetics of off-lattice chains with fixed bond lengths and bond angles and with prescribed sequences. The present treatment is a general purpose one applicable to all macromolecular conformational relaxation processes (e.g., protein folding kinetics, drug/ligand docking on to target proteins, conformational multiple-minima problems, etc.). It serves as a complement to the molecular dynamics or Brownian dynamics treatments in atomic coordinate space.