Time-stepping Solution Research Articles

Forced vibration characteristics of viscoelastic variable stiffness laminated composite plates using time and frequency domain approaches

The linear forced vibration characteristics of viscoelastic variable stiffness laminated composite plates (VVSLC) are studied using a generalized Maxwell model and finite element method. The integral form of the viscoelastic constitutive relation is converted to the incremental form for finite element formulation based on the Reissner–Mindlin plate theory. The recursive relations are developed to compute the current time-step solution using only the previous time-step solution. The periodic response directly in the time domain is obtained using shooting technique coupled with Newmark’s time integration method. The implementation of shooting technique for Boltzmann integral-based viscoelasticity for curvilinear fibre composite plates is done for the first time in this study. For the comparison purpose, the response/resonance frequency/modal loss factor is also obtained using equivalent complex modulus based viscoelastic correspondence principle. It is observed that the variation in fibre orientation and boundary conditions leads to significant variations in response, stress/moment resultant amplitude and the damping factor of the VVSLC plates. Further, the present time domain based approach is capable of predicting damping factor at all forcing frequencies whereas complex eigenvalue analysis can predict damping factor only at discrete resonance frequency. Based on the detailed studies, it is found that the curvilinear fibre composite plates depict a significant reduction in response/moment resultants compared to straight fibre composite plates.

F-PICNN: A physics-informed convolutional neural network for partial differential equations with space-time domain

The physics and interdisciplinary problems in science and engineering are mainly described as partial differential equations (PDEs). Recently, a novel method using physics-informed neural networks (PINNs) to solve PDEs by employing deep neural networks with physical constraints as data-driven models has been pioneered for surrogate modelling and inverse problems. However, the original PINNs based on fully connected neural networks pose intrinsic limitations and poor performance for the PDEs with nonlinearity, drastic gradients, multiscale characteristics or high dimensionality in which the complex features are hard to capture. This leads to difficulties in convergence to correct solutions and high computational costs. To address the above problems, in this paper, a novel physics-informed convolutional neural network framework based on finite discretization schemes with a stack of a series of nonlinear convolutional units (NCUs) for solving PDEs in the space-time domain without any labelled data (f-PICNN) is proposed, in which the memory mechanism can considerably speed up the convergence. Specifically, the initial conditions (ICs) are hard-encoded into the network as the first time-step solution and used to extrapolate the next time-step solution. The Dirichlet boundary conditions (BCs) are constrained by soft BC enforcement while the Neumann BCs are hard enforced. Furthermore, the loss function is designed as a set of discretized PDE residuals and optimized to conform to physics laws. Finally, the proposed auto-regressive model has been proven to be effective in a wide range of 1D and 2D nonlinear PDEs in both space and time under different finite discretization schemes (e.g., Euler, Crank Nicolson and fourth-order Runge-Kutta). The numerical results demonstrate that the proposed framework not only shows the ability to learn the PDEs efficiently but also provides an opportunity for greater conceptual simplicity, and potential for extrapolation from learning the PDEs using a limited dataset.

Methods for Determining Losses and Parameters of Cylindrical-Rotor Medium-Power Synchronous Generators

This paper presents the analytical and numerical application of a method to determine the parameters and power losses in the core of two medium-power synchronous generators. These generators are used as emergency power sources powered by diesel engines, gas engines, and gas turbines. They cover peak electricity demand but can also be used in traction drives. This article presents a new numerical method for determining losses in the generator core based on the use of a time-stepping solution using the FEM method and calculating these losses using analytical formulas. In calculating the losses for the FEM method, approximations of the loss characteristics of the sheet were used with a wide range of induction values and frequencies. This method is specific to the solution used and was adapted from the authors’ previous work on losses in induction machines. A one-phase winding with alternating voltage was supplied to determine the basic parameters in the form of synchronous reactance. Also, an important novelty is the introduction of a new method of determining the saturation state of the magnetic circuit, which significantly affects the machine parameters. The obtained results were used in analytical calculations and implemented in a computer program that allows for the calculation of electromagnetic parameters, operating characteristics, and core losses, taking into account additional losses, total losses, and efficiency, as well as machine parameters in unsteady operating states and the current characteristics of a three-phase symmetrical short circuit at the machine terminals. The calculations obtained were verified experimentally by measurements of real machines.

Nonlinear periodic response of viscoelastic laminated composite plates using shooting technique

The nonlinear periodic response of viscoelastic laminated composite plates subjected to harmonic excitation is carried out in time domain using the generalized Maxwell model in the form of a Boltzmann integral. The integral form of the viscoelastic constitutive relation is converted to the incremental form for the finite element formulation based on the Reissner–Mindlin plate theory incorporating von Karman geometric nonlinearity. The recursive relations are developed to compute the current time-step solution using only the previous time-step solution. The periodic response is obtained using shooting technique coupled with Newmark’s time integration and arc length continuation method. The implementation of the shooting technique for the Boltzmann Integral based viscoelasticity done for the first time in this study involves derivation of a number of additional recursive relations and initial conditions at the beginning of each shooting cycle. The nonlinear periodic vibration characteristics such as frequency response, damping factor, steady-state response history, phase plane plots, and frequency spectra are presented for viscoelastic plates with different boundary conditions and lamination schemes. Further, the influence of relaxation time and number of Maxwell elements on the frequency response characteristics are investigated. It is seen that the shooting technique is capable of predicting periodic responses of viscoelastic dynamical systems directly from the second-order equations of motion and is more efficient than the direct time integration method. The nonlinear response of viscoelastic plates predicted using the equivalent elastic model with Rayleigh proportional damping (having same linear frequency response) is found to be significantly different from the one predicted using viscoelastic constitutive model.

Deep convolutional architectures for extrapolative forecasts in time-dependent flow problems

Physical systems whose dynamics are governed by partial differential equations (PDEs) find numerous applications in science and engineering. The process of obtaining the solution from such PDEs may be computationally expensive for large-scale and parameterized problems. In this work, deep learning techniques developed especially for time-series forecasts, such as LSTM and TCN, or for spatial-feature extraction such as CNN, are employed to model the system dynamics for advection-dominated problems. This paper proposes a Convolutional Autoencoder(CAE) model for compression and a CNN future-step predictor for forecasting. These models take as input a sequence of high-fidelity vector solutions for consecutive time steps obtained from the PDEs and forecast the solutions for the subsequent time steps using auto-regression; thereby reducing the computation time and power needed to obtain such high-fidelity solutions. Non-intrusive reduced-order modeling techniques such as deep auto-encoder networks are utilized to compress the high-fidelity snapshots before feeding them as input to the forecasting models in order to reduce the complexity and the required computations in the online and offline stages. The models are tested on numerical benchmarks (1D Burgers’ equation and Stoker’s dam-break problem) to assess the long-term prediction accuracy, even outside the training domain (i.e. extrapolation). The most accurate model is then used to model a hypothetical dam break in a river with complex 2D bathymetry. The proposed CNN future-step predictor revealed much more accurate forecasting than LSTM and TCN in the considered spatiotemporal problems.

A nonlinear solver based on residual network for seepage equation

In reservoir simulation, the linearization process after the discretization of the seepage equation will additionally introduce new errors, which can be avoided by using a nonlinear solver based on a deep neural network. In this paper, we propose a new solution method without needing linearization of the discretized seepage equation, which combines residual network and correction iteration method to solve nonlinear system of equations instead of using Newton method. The residual network is trained based on label data generated by seepage equation and approximates the solution, and the error of approximated solution is used as the new input of the trained residual network model for the next correction iteration. The single-phase seepage equation is solved to confirm the effectiveness of the proposed method. Numerical results indicate that, error caused by linearization significantly affects the solution accuracy, which reduces solution accuracy and time step. Under the same time step, the proposed solution method has higher accuracy, and under the premise of ensuring the solution accuracy, the time step of proposed method increases by more than three times.

Recurrent solutions of the Lorenz system of differential equations

By introducing new variables in the systems of difference equations from the paper [B. Zlatanovska and D. Dimovski, Systems of difference equations approximating the Lorentz system of differential equations, Contrib. Sec. Math. Tech. Sci. 33 (2012) 75–96], new systems of difference equations are obtained. The solutions of these new systems of difference equations depend only on the initial values [Formula: see text] and the coefficients [Formula: see text] . The power series whose coefficients are these solutions present the recurrent solutions for the Lorenz system of differential equations. A comparison between the behavior of the Lorenz system of differential equations and the local behavior of its recurrent solutions for small time steps is examined in examples by computer simulations, as in the papers [B. Zlatanovska and D. Dimovski, Systems of difference equations approximating the Lorentz system of differential equations, Contrib. Sec. Math. Tech. Sci. 33 (2012) 75–96; B. Zlatanovska and D. Dimovski, Systems of difference equations as a model for the Lorentz system, in Proc. Fifth Int. Scientific Conf. FMNS - 2013, Vol. I (Blagoevgrad, Bulgaria, 2013), pp. 102–107 and B. Zlatanovska, Approximation for the solutions of Lorenz system with systems of differential equations, Bull. Math. 41(1) (2017) 51–61].

Reduced order modeling of the elliptic systems of the frequency-domain method and its error estimation

This study focuses on reduced-order modeling (ROM) for solving parabolic partial differential equations by applying the Fourier transform in the time variable. Each time-step solution is represented by a combination of Fourier coefficients that consist of the solutions of the frequency-dependent elliptic systems. To recover the time-space approximations, we use the Gaussian quadrature rule based on Legendre-Gauss-Lobatto points for the discrete inversion of the Fourier transformed approximate solutions. In this study, in order to extend the scope of ROM and its utility to the frequency-domain (FD) methodologies, we propose a ROM, based on the finite element approximation, by employing proper orthogonal decomposition to the frequency-dependent elliptic solutions delivered in the process of the FD method. We have verified the performance of our ROM by analyzing its errors and provide the relevant experimental results.

A spatio-temporal adaptive phase-field fracture method

We present an energy-preserving mechanic formulation for dynamic quasi-brittle fracture in an Eulerian–Lagrangian formulation, where a second-order phase-field equation controls the damage evolution. The numerical formulation adapts in space and time to bound the errors, solving the mesh-bias issues these models typically suffer. The time-step adaptivity estimates the temporal truncation error of the partial differential equation that governs the solid equilibrium. The second-order generalized-α time-marching scheme evolves the dynamic system. We estimate the temporal error by extrapolating a first-order approximation of the present time-step solution using previous ones with backward difference formulas; the estimate compares the extrapolation with the time-marching solution. We use an adaptive scheme built on a residual minimization formulation in space. We estimate the spatial error by enriching the discretization with elemental bubbles; then, we localize an error indicator norm to guide the mesh refinement as the fracture propagates. The combined space and time adaptivity allows us to use low-order linear elements in problems involving complex stress paths. We efficiently and robustly use low-order spatial discretizations while avoiding mesh bias in structured and unstructured meshes. We demonstrate the method’s efficiency with numerical experiments that feature dynamic crack branching, where the capacity of the adaptive space–time scheme is apparent. The adaptive method delivers accurate and reproducible crack paths on meshes with fewer elements.

Linking Time-Domain Vibration Behaviors to Spatial-Domain Propagating Waves in a Leaf-like Gradient-Index Phononic Crystal Lens

It is known that a propagating wave at a certain spatial point can be decomposed into plane waves propagating at different angles. In this work, by designing a gradient index phononic crystal lens (GRIN PCL) with transverse-continuous leaf-like unit cells, we theoretically and experimentally show that the spatial-domain propagating waves in finite periodic structures can be linked to their time-domain vibration behaviors. The full-field instantaneous focusing behaviors of Lamb waves in the proposed leaf-like GRIN PCL give an example of the wave-vibration linkage in finite periodic structures while allowing a certain complexity. The conclusion in this paper can help one skip iterative time-consuming finite element analysis (e.g., time-stepping solutions) to avoid possible numerical instabilities occurred in calculating transient wave field on practical finite metamaterials or phononic crystals having unit cells with complicated configurations.

Implicit Space-Time Domain Decomposition Approach for Solving Multiphase Miscible Flow: Accuracy and Scalability

Summary In this paper, we propose a fully implicit space-time multiscale scheme to improve computational efficiency in solving nonlinear multiphase flow in porous media. Here, error estimators are used for adaptively changing the spatial-temporal mesh. This algorithm applied to the black-oil model is compared to a standard control volume approach using a fine time and spatial mesh. Error estimators are introduced to determine subdomains of the reservoir in which high nonlinearity hinders Newtonian convergence. This is followed by applying local fine timesteps to these marked regions, whereas the remaining regions retain the coarse time scale. Once a temporal discretization is determined for different parts of the reservoir, the spatial mesh is refined for treating saturation fronts. The nonmatching interfaces arising from different temporal and spatial scales are resolved by the enhanced velocity method, which enforces strongly the continuity of fluxes. This whole system is solved monolithically. Results from a three-phase black-oil model are described. The multiscale solution is compared to a uniformly fine spatial mesh and fine timestepping solution to confirm accuracy. Solutions from both Gaussian and channelized permeability fields are presented. In the multiscale solution, we observe temporal refinements being applied to the water and gas saturation fronts by reducing the timestep size to guarantee Newtonian convergence. We also observe that the extent of refinements is balanced between the two saturation fronts based on their respective nonlinearity. For Gaussian permeability fields, the algorithm only treats saturation fronts spatially, whereas for channelized permeability fields, the geological features are also considered. Production profiles for the three phases match well between the two solutions under specific refinement criteria. We also investigate the improvement in computational efficiency, as well as the algorithm scalability in regard to increasing problem sizes. We observe a speedup of approximately 10 for solving the linear system, and such speedup increases as the problem size expands. In reservoir simulation, schemes that attempt to decouple the diffusion and advection process become less robust as the physics becomes more complex. Our approach, which is focused on handling the high nonlinearities, can be treated fully implicitly and provide a reduction in simulation costs. This approach can also be applied to model order reduction in providing accurate snapshot solutions. NOTE: This paper is published as part of the 2021 SPE Reservoir Simulation Conference Special Issue.

Implementation of Rayleigh Damping for Local Nonlinear Dynamic Analysis Based on a Matrix Perturbation Approach

Nonlinear analyses of structures under dynamic excitation are becoming increasingly important in structural design and performance evaluation, but large computational effort is the main factor that limits their application. Because nonlinearity is commonly confined within small regions, many studies have been devoted to improving the efficiency of solving such local nonlinear problems by maintaining the structural stiffness elasticity and simulating the effects of local nonlinearity through fictitious nonlinear forces or local modification of the elastic structural response. For dynamic analysis, the nature of the stiffness matrix determines the formulation of the widely used Rayleigh damping; as a result, these local nonlinear analysis methods often use elastic stiffness-based Rayleigh damping models. However, such a damping model can generate unexpected artificial damping forces for inelastic systems and consequently produce inaccurate or even invalid results. Although a tangent stiffness–based Rayleigh damping model has been proven to be a reasonable basis for performing highly accurate dynamic analyses, this type of damping model has difficulty achieving direct compatibility with local nonlinear analysis methods. The present research focuses on the implementation of a tangent stiffness–based Rayleigh damping model for use in efficient local nonlinear analysis methods developed based on the Woodbury formula, which can calculate the structural inelastic behavior by updating the elastic solution rather than by updating the stiffness. By representing tangent stiffness–based Rayleigh damping as a low-rank perturbation to the elastic stiffness-based damping matrix, this study derives a modified dynamic Woodbury formula in which an additional influence coefficient is introduced to reflect the effects of local nonlinearity on the structural damping properties. Moreover, to overcome the potential solution difficulty caused by abrupt changes in the damping forces in certain steps and further improve the efficiency of the proposed method, a variable time step solution scheme that can meet the computational requirements of the Woodbury formula is presented. Verifications demonstrate the validity and high efficiency of the proposed method.

A MOOD-MUSCL Hybrid Formulation for the Non-conservative Shallow-Water System

The stability of the high-order finite volume method for hyperbolic systems is based on non-linear procedures that prevent the creation of non-physical oscillations. Traditional techniques use the a priori paradigm where the procedure is carried out with the current time step solution. The a posteriori paradigm lies on an advanced in time candidate solution and a posterior evaluation of its stability, followed by a cure when necessary. To compare the two strategies, we propose a detailed study using the non-conservative shallow-water equations as a prototype, where both techniques are applied together in the framework of second-order linear reconstruction for the sake of simplicity. Then, a hybrid version combining the most positive aspects of both methods is proposed and analysed.

Isogeometric residual minimization (iGRM) for non-stationary Stokes and Navier–Stokes problems

We show that it is possible to obtain a linear computational cost FEM-based solver for non-stationary Stokes and Navier–Stokes equations. Our method employs a technique developed by Guermond and Minev (2011), which consists of singular perturbation plus a splitting scheme. While the time-integration schemes are implicit, we use finite elements to discretize the spatial counterparts. At each time-step, we solve a PDE having weak-derivatives in one direction only (which allows for the linear computational cost), at the expense of handling strong second-order derivatives of the previous time step solution, on the right-hand side of these PDEs. This motivates the use of smooth functions such as B-splines. For high Reynolds numbers, some of these PDEs become unstable. To deal robustly with these instabilities, we propose to use a residual minimization technique. We test our method on problems having manufactured solutions, as well as on the cavity flow problem.

An asynchronous MATE-multirate method for the modeling of electric power systems

The multiarea Thevenin equivalent (MATE) solution framework is used in this work to combine several subsystems, which are solved using different multiple and non-multiple time-step sizes. In the proposed asynchronous MATE-multirate (A-MATE-multirate) solution, each subsystem is solved with the integration step size that is most adequate for its own internal time constants. This paper relaxes the requirement of previous multirate techniques, in which the solution time-steps of different systems must be multiples or submultiples of each other. To this end, this paper introduces the concept of asynchronous updating, in which each subsystem is updated according to its own internal clock. This updating process is performed in a closed-form solution, with external subsystems represented by their interpolated Thevenin equivalents. Interpolation with future values without using extrapolation is possible in the electromagnetic transients program solution because the history source for future time-steps is calculated at previous time-steps. The exchange of information between subsystems is conditioned using filters to avoid numerical problems during the decimation of the fast subsystems and the interpolation of the slow subsystems. The effectiveness of the proposed methodology is verified with examples using multiple and non-multiple time-step sizes.

A Hierarchical Low-Rank Approximation Based Network Solver for EMT Simulation

In electromagnetic transient (EMT) simulation, 80–97% of the computational time is devoted to solving the network equations. A key observation is that the sub-matrix representing the interaction between two far-away groups of buses is usually sparse and can be approximated by a low-rank matrix. Based on this observation, we propose a novel low-rank approximation method which permits ${O(N \log N)}$ -time matrix-vector multiplication for each network solution time step. Comprehensive numerical studies are conducted on a 39-bus system and a 179-bus system from the literature, and large cases created from the two systems. The results demonstrate that the proposed approach is up to $2.8\times$ faster than the state-of-the-art sparse LU factorization based network solution, without compromising simulation accuracy. Since our low-rank approximation is highly parallelizable, further speedup may be possible.

Modeling Arbitrarily Shaped Liquid Pipelines Using a Segmented Transmission Line Model

Abstract This paper presents a new method of simulating the dynamic flow and pressure of laminar liquid flow through pipes of arbitrarily changing cross section. This method uses a segmented model based on the previously presented tapered transmission line model (TLM). This new method is computationally efficient and has comparable accuracy to previous methods such as the method of characteristics (MOC), but allow for more flexibility in solution time-step (such as accommodating variable time-step solvers), which is required if the rest of the system model has stiff equations. For the sample geometry presented, the new model calculates the dynamic response an order of magnitude faster than the previous method of characteristics solution, with minimal loss of accuracy.

Analytical time-stepping solution of the discretized population balance equation

The prediction of the particle-size distribution (PSD) of the particulate systems in chemical engineering is very important in a variety of different contexts, such as parameter identification, troubleshooting, process control, design, product quality, production economics etc. The time evolution of the PSD can be evaluated by means of the population balance equation (PBE), which is a complex integro-differential equation, whose solution in practical cases always requires sophisticated numerical methods that may be computationally tedious. In this work, we propose a novel technique that tackles this issue by using an analytical time-stepping procedure (ATS) to resolve the PSD time dependency. The ATS is an explicit time integrator, taking advantage of the linear or almost linear time dependency of the discretized population balance equation. Thus, linear approximation of the source term is a precondition for the ATS simulations. The presented technique is compared with a standard variable step time integrator (MATLAB ODE15s stiff solver), for practical examples e.g. emulsion, aging cellulose process, cooling crystallization, reactive dissolution, and liquid-liquid extraction. The results show that this advancing in time procedure is accurate for all tested practical examples, allowing reproducing the same results given by standard time integrators in a fraction of the computational time.

An analysis on the influence of air ingestion on vibration damping properties of squeeze film dampers

Considering the effect of air ingestion, a three-dimensional CFD model of open ends squeeze film damper (SFD) was established based on transient time step solution method and dynamic mesh technique. And the model was experimentally verified by bidirectional excitation experiment. Experiment and CFD simulation shows that air ingestion was caused by the pressure difference between oil film and external environment. With increasing precession frequency and eccentricity ratio, air ingestion area is gradually expanding resulting in increasing air volume, shrinking effective coverage of oil film and performance degradation of damper. Thereby, oil film damping shows the tendency to reduce. Compared with the case without considering air ingestion, the case considering air ingestion has obviously smaller peak pressure in the CFD simulation.

Hp-version time domain boundary elements for the wave equation on quasi-uniform meshes

Solutions to the wave equation in the exterior of a polyhedral domain or a screen in R3 exhibit singular behavior from the edges and corners. We present quasi-optimal hp-explicit estimates for the approximation of the Dirichlet and Neumann traces of these solutions for uniform time steps and (globally) quasi-uniform meshes on the boundary. The results are applied to an hp-version of the time domain boundary element method. Numerical examples confirm the theoretical results for the Dirichlet problem both for screens and polyhedral domains.