Accurate Time Integration Research Articles

Enhancing energy capacity of the car’s hood door under excitation frequency via functionally graded triply periodic minimal surface material: Application of machine learning and mathematical simulation

This study investigates the nonlinear dynamic response and vibration characteristics of a car’s hood door, focusing on its energy capacity under various excitation frequencies. The hood door is modeled using a functionally graded triply periodic minimal surface (FG-TPMS) material, which offers superior mechanical properties and lightweight design. The analysis is conducted using a higher-order shear deformation theory (HSDT), which accounts for shear deformations more accurately than classical theories. The incorporation of von Karman nonlinear terms captures the geometric nonlinearity due to large deformations, providing a realistic simulation of the hood door’s behavior under dynamic loads. To solve the complex equations of motion derived from the HSDT and von Karman terms, a fourth-order Runge–Kutta method is employed. This numerical method ensures accurate time integration and stability of the solution. The dynamic response is further analyzed to determine the energy absorption capacity of the hood door material, crucial for safety and durability in automotive applications. Validation of the numerical model is performed using previous published articles and a hybrid machine learning algorithm, which combines data-driven approaches with traditional physics-based models. This hybrid validation enhances the accuracy and reliability of the predicted responses, ensuring that the proposed model can be effectively used in the design and optimization of car components. The results demonstrate the potential of FG-TPMS materials in automotive applications, offering improved energy absorption and vibration control, which are essential for enhancing vehicle safety and performance.

Effective highly accurate time integrators for linear Klein–Gordon equations across the scales

Effective highly accurate time integrators for linear Klein–Gordon equations across the scales

One-sweep moment-based semi-implicit-explicit integration for gray thermal radiation transport

Thermal radiation transport (TRT) is a time dependent, high dimensional partial integro-differential equation. In practical applications such as inertial confinement fusion, TRT is coupled to other physics such as hydrodynamics, plasmas, etc., and the timescales one is interested in capturing are often much slower than the radiation timescale. As a result, TRT is treated implicitly, and due to its stiffness and high dimensionality, is often a dominant computational cost in multiphysics simulations. Here we develop a new approach for implicit-explicit (IMEX) integration of gray TRT in the deterministic SN setting, which requires only one sweep per stage, with the simplest first-order method requiring only one sweep per time step. The partitioning of equations is done via a moment-based high-order low-order formulation of TRT, where the streaming operator and first two moments are used to capture the asymptotic stiff regimes of the streaming limit and diffusion limit. Absorption-reemission is treated explicitly, and although stiff, is sufficiently damped by the implicit solve that we achieve stable accurate time integration without incorporating the coupling of the high order and low order equations implicitly. Due to nonlinear coupling of the high-order and low-order equations through temperature-dependent opacities, to facilitate IMEX partitioning and higher-order methods, we use a semi-implicit integration approach amenable to nonlinear partitions. Results are demonstrated on thick Marshak and crooked pipe benchmark problems, demonstrating orders of magnitude improvement in accuracy and wallclock compared with the standard first-order implicit integration typically used.

Structure-Preserving Discretization of a Polyconvexity-Inspired Formulation for Coupled Nonlinear Electro-Thermo-Elastodynamics

A consistent, structure-preserving space-time discretization for coupled nonlinear electro-thermo-elastodynamical problems is presented. The underlying polyconvexity-inspired mixed framework is facilitated by the properties of the tensor cross product. The elastodynamic problem is then extended by the energy balance as well as Gauss’s and Faraday’s law to integrate the thermodynamic and electrostatic contribution, respectively. A suitable polyconvexity-inspired internal energy function is chosen to complete the nonlinear, fully coupled electro-thermo-elastodynamical formulation. Additionally, we present a structure-preserving, second-order accurate time integration scheme, utilizing discrete derivatives in the sense of Gonzalez (1996), ensuring a stable and robust simulation even for large time steps. Finally, we assess the numerical performance of our newly developed method through representative examples also showing the possibilities of the framework in the field of boundary control.

High-order relaxation methods for nonequilibrium two-phase flow equations

PurposeThe purpose of this paper is to investigate the two-phase flow problems involving gas–liquid mixture.Design/methodology/approachThe governed equations consist of a range of conservation laws modeling a classification of two-phase flow phenomena subjected to a velocity nonequilibrium for the gas–liquid mixture. Effects of the relative velocity are accounted for in the present model by a kinetic constitutive relation coupled to a collection of specific equations governing mass and volume fractions for the gas phase. Unlike many two-phase models, the considered system is fully hyperbolic and fully conservative. The suggested relaxation approach switches a nonlinear hyperbolic system into a semilinear model that includes a source relaxation term and characteristic linear properties. Notably, this model can be solved numerically without the use of Riemann solvers or linear iterations. For accurate time integration, a high-resolution spatial reconstruction and a Runge–Kutta scheme with decreasing total variation are used to discretize the relaxation system.FindingsThe method is used in addressing various nonequilibrium two-phase flow problems, accompanied by a comparative study of different reconstructions. The numerical results demonstrate the suggested relaxation method’s high-resolution capabilities, affirming its proficiency in delivering accurate simulations for flow regimes characterized by strong shocks.Originality/valueWhile relaxation methods exhibit notable performance and competitive features, as far as we are aware, there has been no endeavor to address nonequilibrium two-phase flow problems using these methods.

An implicit-explicit solver for a two-fluid single-temperature model

We present an implicit-explicit finite volume scheme for two-fluid single-temperature flow in all Mach number regimes which is based on a symmetric hyperbolic thermodynamically compatible description of the fluid flow. The scheme is stable for large time steps controlled by the interface transport and is computational efficient due to a linear implicit character. The latter is achieved by linearizing along constant reference states given by the asymptotic analysis of the single-temperature model. Thus, the use of a stiffly accurate IMEX Runge Kutta time integration and the centered treatment of pressure based quantities provably guarantee the asymptotic preserving property of the scheme for weakly compressible Euler equations with variable volume fraction. The properties of the first and second order scheme are validated by several numerical test cases.

Fast parallel IGA-ADS solver for time-dependent Maxwell's equations

We propose a simulator for time-dependent Maxwell's equations with linear computational cost. We employ B-spline basis functions as considered in the isogeometric analysis (IGA). We focus on non-stationary Maxwell's equations defined on a regular patch of elements. We employ the idea of alternating-directions splitting (ADS) and employ a second-order accurate time-integration scheme for the time-dependent Maxwell's equations in a weak form. After discretization, the resulting stiffness matrix exhibits a Kronecker product structure. Thus, it enables linear computational cost LU factorization. Additionally, we derive a formulation for absorbing boundary conditions (ABCs) suitable for direction splitting. We perform numerical simulations of the scattering problem (traveling pulse wave) to verify the ABC. We simulate the radiation of electromagnetic (EM) waves from the dipole antenna. We verify the order of the time integration scheme using a manufactured solution problem. We then simulate magnetotelluric measurements. Our simulator is implemented in a shared memory parallel machine, with the GALOIS library supporting the parallelization. We illustrate the parallel efficiency with strong and weak scalability tests corresponding to non-stationary Maxwell simulations.

Effect of temperature on convective-reactive transport of CO2 in geological formations

Geological CO2 sequestration (GCS) remains the main promising solution to mitigate global warming. Understating the fate of CO2 behavior is crucial for securing its containment in the reservoir and predicting the impact of dissolved CO2 on the host formation. Most modeling-based studies in the literature investigated the convective-reactive transport of CO2 by assuming isothermal conditions. The effect of temperature on the convective-reactive transport of CO2 is still poorly understood, particularly at the field scale. The objective of this study is to provide an in-depth understanding of CO2-related reactive thermohaline convection (RTHC) processes at field scale. Thus, a new numerical model based on advanced finite element formulations is developed. The new model incorporates an accurate time integration scheme with error control. Numerical experiments confirm high accuracy and efficiency of the newly developed model. The effect of temperature on CO2 transport is investigated for a field case in the Viking reservoir in the North Sea. Results show that including the temperature effect intensifies the fingering processes and, consequently, CO2 dissolution. Neglecting the thermal convection processes and the impact of temperature on the dissolution rate can significantly impact the model predictions. A sensitivity analysis is developed to understand the effect of parameters governing the dissolution rate on the fingering phenomenon and the total CO2 flux.

Higher-Order Accurate Explicit Time Schemes with Improved Dissipation Properties

In this paper, a novel family of fourth-order accurate explicit time integration schemes is developed by combining the new time approximations and the explicit fourth-order Runge–Kutta (RK4) method. Inaccurate predictions due to the presence of excessive numerical dissipation are often observed in practical analyses of structural dynamics when the RK4 is employed for both displacement and velocity approximations. To remedy this, novel time approximations with adjustable algorithmic parameters are employed for the displacement vectors while the velocity vectors are approximated by using the RK4. For the complete elimination of numerical dissipation, algorithmic parameters are unconventionally determined by taking the determinant of the amplification matrix as unity. A set of algorithmic parameters obtained from this process makes the new schemes completely non-dissipative while keeping the computational cost the same as the RK4. Due to this improvement, the new schemes have enhanced total energy-conserving capabilities for nonlinear systems and give noticeably more accurate predictions in practical analyses. Until now, controllable numerical dissipation and fourth-order accuracy are not attained simultaneously in a unified set of time schemes. In the new schemes, however, a systematic way to adjust the level of numerical dissipation is also presented while attaining fourth-order accuracy. The numerical results of various test problems show that the new time schemes can provide more accurate predictions for nonlinear conservative dynamic problems than the existing time schemes.

An Accurate, Controllably Dissipative, Unconditionally Stable Three-Sub-Step Method for Nonlinear Dynamic Analysis of Structures

An implicit truly self-starting time integration method for nonlinear structural dynamical systems is developed in this paper. The proposed method possesses unconditional stability, second-order accuracy, and controllable dissipation, and it has no overshoots. The well-known BN-stability theory is employed in the design of algorithmic parameters, ensuring that the proposed method can stably solve nonlinear structural dynamical systems without restricting the time step size. The spectral analysis shows that compared to existing second-order accurate time integration methods, the proposed method enjoys a considerable advantage in low-frequency accuracy. For nonlinear problems where the currently popular Generalized-[Formula: see text] method and [Formula: see text]-Bathe method fail, the proposed method shows strong stability and accuracy. Further, for nonlinear problems in which all methods’ results are convergent, the proposed method has greater accuracy, efficiency, and energy-conservation capability.

Energy Preserving Accurate Time Integration for the Vlasov Equation

Energy Preserving Accurate Time Integration for the Vlasov Equation

Energy Preserving Accurate Time Integration for the Vlasov Equation

Energy Preserving Accurate Time Integration for the Vlasov Equation

Highly Accurate and Efficient Time Integration Methods with Unconditional Stability and Flexible Numerical Dissipation

This paper constructs highly accurate and efficient time integration methods for the solution of transient problems. The motion equations of transient problems can be described by the first-order ordinary differential equations, in which the right-hand side is decomposed into two parts, a linear part and a nonlinear part. In the proposed methods of different orders, the responses of the linear part at the previous step are transferred by the generalized Padé approximations, and the nonlinear part’s responses of the previous step are approximated by the Gauss–Legendre quadrature together with the explicit Runge–Kutta method, where the explicit Runge–Kutta method is used to calculate function values at quadrature points. For reducing computations and rounding errors, the 2m algorithm and the method of storing an incremental matrix are employed in the calculation of the generalized Padé approximations. The proposed methods can achieve higher-order accuracy, unconditional stability, flexible dissipation, and zero-order overshoots. For linear transient problems, the accuracy of the proposed methods can reach 10−16 (computer precision), and they enjoy advantages both in accuracy and efficiency compared with some well-known explicit Runge–Kutta methods, linear multi-step methods, and composite methods in solving nonlinear problems.

A time-domain POD approach based on numerical implicit and explicit Green’s functions for 3D elastodynamic analysis

This paper is concerned with the development of time integration methods based on numerical Green’s functions applied in conjunction with the proper orthogonal decomposition (POD) technique to solve three-dimensional elastodynamic problems discretized by the FEM. The convolution integral in the velocity expression of these methods is firstly carried out through integration by parts, and then approximated numerically, resulting in a simpler and more accurate convolution than that reported in the literature. To compute the solution samples, required by the POD technique, the Implicit Green’s functions Approach (ImGA)-Newmark method rewritten in terms of the ultimate spectral radius is employed. The proposed ImGA scheme is truly self-starting and easy to implement with just one free parameter. To solve the reduced-order model, a new highly accurate time integration scheme based on the Explicit Green’s functions Approach (ExGA)-Runge–Kutta method is proposed, considering the Gauss–Lobatto-Legendre (GLL) points to perform both the time integration of the Green’s functions and the approximation of the convolution integrals. To verify the accuracy and efficiency of the proposed formulation, the numerical results are compared with those from the full-order model. Moreover, undamped solution samples are utilized to build a reduced-order model that can take into account a Rayleigh damping matrix.

A novel time‐domain numerical methodology for the electromagnetic analysis of an H‐plane tee power divider

A time-domain numerical methodology for the transient analysis of multisection waveguide power dividers is proposed. Waveguide power dividers are essential components of microwave systems for high-power applications. A numerical strategy based on the 3D discontinuous Galerkin time-domain finite element method is used to estimate the electromagnetic propagation inside a power divider with three ports. The proposed approach leads to simulations that are fast, robust, and accurate, this being the result of a parallel local discretisation of the full-wave time-domain Maxwell's equations on unstructured tetrahedral meshes. The stability of the overall numerical scheme is obtained using an upwind numerical flux, an implicit second-order accurate time integration scheme, and Whitney's edge elements. Good agreements are obtained when the computed results are compared with other HFSS simulation results, and with measurement results found in the literature.

Assessment of an adaptive time integration strategy for a high‐order discretization of the unsteady RANS equations

SummaryMany industrial applications, for example, aeronautics, aeroacoustics, and turbomachinery, are characterized by complex turbulent flow problems, whose numerical studies are mostly based on statistical models. These models rely on the Navier–Stokes equations averaged in time, namely the Reynolds‐averaged Navier–Stokes (RANS) equations. RANS unsteady solutions require accurate and efficient time integration strategies, and the choice of the time‐step can have a strong impact on the robustness and efficiency of the simulation. Adaptive time step algorithms can improve considerably the effectiveness of time integration, but in literature little information is available on their application to unsteady RANS equations for high‐Reynolds number turbulent flows. This work aims at reducing this gap, presenting a numerical investigation of the performance of different adaptive temporal strategies applied to linearly implicit Rosenbrock‐type schemes within a high‐order discontinuous Galerkin framework. Several test cases of increasing stiffness and difficulty are considered for both compressible and incompressible turbulent flows.

A mixed finite element formulation for energy‐momentum time integrations of composites with fiber bending stiffness

AbstractWe aim at dynamic simulations of 3D‐fiber‐reinforced materials in light‐weight structures, which are reinforced by fiber rovings comprised of thousands of filaments. Each roving possesses a separate bending stiffness, which has a large influence on the material behaviour on the macro scale. For the description of this material behavior, we extend a hyperelastic anisotropic Cauchy continuum formulation, which models only the macro‐scopic behaviour of a reinforcement by filaments, by a higher‐order gradient of the deformation mapping as an independent field. The new mixed finite element formulation is based on well‐known mixed finite elements for anisotropic polyconvex material formulations, but include new additional mixed fields concernd with the fiber bending stiffness. As we aim at dynamic long‐term simulations, we apply higher‐order accurate time integrators. Here, we include the new mixed finite element formulation in a variational‐based time finite element method. As numerical examples serve a simple cantilever beam. Here, we first analyze the influence of the fiber bending stiffness.

On accurate time integration for temperature evolutions in additive manufacturing

AbstractWe investigate two numerical challenges in thermal finite element simulations of laser powder bed fusion (LPBF) processes. First, we compare the behavior of first‐ and second‐order implicit time‐stepping schemes on a fixed domain. While both methods yield comparable accuracies in the pre‐asymptotic regime, the second‐order method eventually outperforms the first‐order method. However, the oscillations present in the pre‐asymptotic range of the second‐order method can render it less suitable for simulating LPBF processes. Then, we consider sudden domain extensions resulting from subsequently adding new layers of material with ambient temperature. We model this extension on the continuous level in an energy conservative manner. The discontinuities introduced here reduce the convergence order for both time‐stepping schemes to 0.75. First and second order accuracy could only be achieved by strongly grading the time‐steps towards the domain expansion.

A unified second-order accurate in time MPM formulation for simulating viscoelastic liquids with phase change

We assume that the viscous forces in any liquid are simultaneously local and non-local , and introduce the extended POM-POM model [McLeish and Larson 1998; Oishi et al. 2012; Verbeeten et al. 2001] to computer graphics to design a unified constitutive model for viscosity that generalizes prior models, such as Oldroyd-B, the Upper-convected Maxwell (UCM) model [Sadeghy et al. 2005], and classical Newtonian viscosity under one umbrella, recovering each of them with different parameter values. Implicit discretization of our model via backward Euler recovers the variational Stokes solver of [Larionov et al. 2017] for Newtonian viscosity. For greater accuracy, however, we introduce the second-order accurate Generalized Single Step Single Solve (GS4) scheme [Tamma et al. 2000; Zhou and Tamma 2004] to computer graphics, which recovers all prior second-order accurate time integration schemes to date. Using GS4 and our generalized constitutive model, we present a Material Point Method (MPM) for simulating various viscoelastic liquid behaviors, such as classical liquid rope coiling, buckling, folding, and shear thinning/thickening. In addition, we show how to couple our viscoelastic liquid simulator with the recently introduced non-Fourier heat diffusion solver [Xue et al. 2020] for simulating problems with phase change, such as melting chocolate and digital fabrication with 3D printing. While the discretization of heat diffusion is slightly different within GS4, we show that it can still be efficiently solved using an assembly-free Multigrid-preconditioned Conjugate Gradients solver. We present end-to-end 3D simulations to demonstrate the versatility of our framework.

A three-time-level a posteriori error estimator for GS4-2 framework: Adaptive time stepping for second-order transient systems

The most commonly used adaptive time stepping technique is mainly based on the simple notion of a local error estimation. Accordingly, a variety of error estimators have mostly been designed for particular algorithms on an algorithm by algorithm selection basis. In this paper, a novel design of a three-time-level general purpose a posteriori error estimator is demonstrated for second-order transient systems, providing features of: (a) being universal for the most popular and entire class of second-order accurate unconditionally stable implicit LMS methods with/without numerical dissipation developed over the past few decades and on newer and optimal designs (the focus is on unconditionally stable implicit LMS methods and not explicit methods); (b) being simple for numerical implementation in the general purpose GS4-2 framework encompassing the entire class of LMS methods that are second order time accurate such that only the acceleration at three time levels is required; (c) preserving the third-order convergence in time for the local error; (d) predicting accurate local error such that the effectivity index is close to unity. In contrast to the existing a posteriori error estimators which are designed for certain algorithms and on algorithm by algorithm basis, the newly proposed approach can be directly applied to all the known and existing and new second-order accurate time integration algorithms in the entire class of the unconditionally stable implicit LMS family under the umbrella of the GS4-2 framework; thereby, the cumbersome work of designing different error estimators for different algorithms is avoided. In particular, the proposed error estimator can be directly applied also to the novel V0 family of algorithms without any limitation. Besides the limited selection of estimators existing within the U0 family of algorithms where most of the traditional methods belong to, the remainder of the algorithms within the U0 family and several others such as the V0 family do not have estimators that can be readily designed as the V0 family of algorithms only exist in the GS4-2 framework, which are entirely novel and new and are highly competitive. The numerical examples of stiff nonlinear spring pendulum, van der Pol equation, wave propagation, and phase-field modeling are described here to show the benefits to the computational cost whilst preserving generality, accuracy, and the like when the adaptive time stepping approach coupled with the proposed error estimator is applied to the transient simulations.