Explicit Time Integration Method Research Articles

An Explicit Time Integration Method Based on the Verlet Scheme with Improved Characteristics in Numerical Dispersion

An Explicit Time Integration Method Based on the Verlet Scheme with Improved Characteristics in Numerical Dispersion

Study of pulse wave phenomena associated with blood flow model in human viscoelastic artery

Cardiac output monitoring has proven to be a promising hemodynamic management tool especially for critically ill patients. Pulse wave analysis is a noninvasive method used to quantify cardiac output continuously with respect to time. In this article, we have proposed a novel methodology to quantify the contribution of pulse waves to further study the role of arterial wall relaxation with respect to time. The relaxation time can further help in the diagnosis of disease. The pulse wave velocity component is derived by transforming governing fluid flow equations into hyperbolic equations for laminar incompressible blood flow in an artery of viscoelastic walls. The viscoelastic behavior of the wall is analogously modeled by the modified Zener model that has the capability to measure creep, stress relaxation, and hysteresis. The derived model equations are solved numerically by the multiderivative Runge–Kutta implicit–explicit time integration method with a weighted, essentially non-oscillatory, discretization scheme. The results are well validated with clinical trial-based Riemann problems for the case of elastic walls. It is observed that the proposed modified Zener model is well suitable to identify the location of arterial stiffness where the pulse wave manifests various types of phenomena like discontinuity and reflection.

Implicit-explicit time integration for the immersed wave equation

Efficient solution strategies for wave propagation problems with complex geometries are essential in many engineering fields. Immersed boundary methods simplify mesh generation by embedding the domain of interest into an extended domain that is easy to mesh, introducing the challenge of dealing with cells that intersect the domain boundary. We consider the finite cell method that extends the weak form from the physical into the extended domain, multiplied by a small value, to stabilize badly cut cells. Combined with explicit time integration methods, the presence of finite cells with very little support in the physical domain results in tiny critical time step sizes. While the finite cell stabilization limits how small the critical time step size can become, this limit is still restrictive for many applications. Explicit transient analyses commonly use the spectral element method due to its natural way of obtaining diagonal mass matrices through nodal lumping. The resulting method is very efficient since nodal lumping renders the solution of the equation systems trivial while not reducing the accuracy or the critical time step size. The combination of the spectral element and finite cell methods is called the spectral cell method. Unfortunately, a direct application of nodal lumping in the spectral cell method is impossible due to the special quadrature necessary to treat the discontinuous integrand inside the cut cells. The existing approaches to lump the mass matrices of cut cells significantly reduce the accuracy of the approximation.We analyze an implicit-explicit (IMEX) time integration method to exploit the advantages of the nodal lumping scheme for uncut cells on one side and the unconditional stability of implicit time integration schemes for cut cells on the other. In this hybrid, immersed Newmark IMEX approach, we use explicit second-order central differences to integrate the uncut degrees of freedom that lead to a diagonal block in the mass matrix and an implicit trapezoidal Newmark method to integrate the remaining degrees of freedom (those supported by at least one cut cell). The immersed Newmark IMEX approach preserves the high-order convergence rates and the geometric flexibility of the finite cell method while retaining the efficiency of the nodal lumping scheme for uncut cells without compromising the critical time step size. We analyze a simple system of spring-coupled masses to highlight some of the essential characteristics of Newmark IMEX time integration. We then solve the scalar wave equation with immersed boundaries on two- and three-dimensional examples with significant geometric complexity to show that our approach is more efficient than state-of-the-art time integration schemes when comparing accuracy and runtime. While we focus on the finite and spectral cell methods, we expect other immersed methods to benefit equally from Newmark IMEX time integration.

Proper Orthogonal Decomposition Based Intrusive Reduced Order Models to Accelerate Computational Speed of Dynamic Analyses of Structures Using Explicit Time Integration Methods

Proper Orthogonal Decomposition Based Intrusive Reduced Order Models to Accelerate Computational Speed of Dynamic Analyses of Structures Using Explicit Time Integration Methods

A Godunov-type stabilization scheme for the Boltzmann transport equation of III-V devices with a 3D k-space

This paper presents a deterministic approach for solving the Boltzmann transport equation (BTE) together with the Poisson equation (PE) for III-V semiconductor devices with a three-dimensional k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf {k}}$$\\end{document}-space. The BTE is stabilized using Godunov’s scheme, whose linearity in the distribution function simplifies the application of the Newton–Raphson method to the coupled discrete BTE and PE. The formulation of the discrete equations ensures the nonnegativity of the distribution function regardless of the scattering rate, which can include the Pauli exclusion principle, and exhibits excellent numerical stability under steady state as well as transient conditions. In the latter case, both implicit and explicit time integration methods can be used and even slow processes (e.g., recombination) can be handled using this approach. In addition, the direct solution of the BTE can be easily extended to the small-signal case for arbitrary frequencies. Exemplary BTE results are shown for a GaAs N+NN+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extrm{N}}^{+}{\ extrm{NN}}^{+}$$\\end{document}-structure, revealing, inter alia, that the approximations of the drift-diffusion model can fail for large built-in fields in III-V devices.

Improved modal analyses using the novel quadrilateral overlapping elements

In this work, the novel quadrilateral overlapping element incorporated with the harmonic trigonometric functions is presented to reduce the efforts in mesh division for modal analysis. The new partition of unity functions in the overlapping element are cubic polynomials. Therefore, low-order polynomials can be utilized to augment local approximation space without the linear dependence problem. Moreover, the original polynomial approximation spaces are further enriched by the harmonic trigonometric functions without introducing additional rank deficiencies in the system matrices. Thus, the presented enriched overlapping finite element method (EOFEM) provides very precise eigenfrequencies and eigenmodes by using a very coarse mesh. However, the shape functions in the presented EOFEM do not possess the Kronecker delta function property. Hence, a conforming method that modifies the shape functions rather than the variational principle is proposed to impose the essential boundary conditions. Several numerical examples in two dimensions are then investigated to elaborate on the performance of the presented EOFEM in modal analyses. Besides, a block-diagonal lumped mass matrix is presented and further extended to the structure vibration analysis by incorporating the explicit time integration method. It is proven that the enriched overlapping element is computationally more efficient than the quadratic finite element and the original overlapping element.

Hermes-3: Multi-component plasma simulations with BOUT++

A new open source tool for fluid simulation of multi-component plasmas is presented, based on a flexible software design that is applicable to scientific simulations in a wide range of fields. Hermes-3 is built on plasma simulation framework BOUT++, consolidating earlier SD1D and Hermes models into a single code that can be configured at run-time to solve plasma models in 1D, 2D or 3D, either for transport (steady-state) or turbulent (time-evolving) problems, with an arbitrary number of ion and neutral species. We describe the improved numerical algorithms and software design that have been implemented in Hermes-3.To demonstrate the capabilities of this tool, applications relevant to the boundary of tokamak plasmas are presented: 1D simulations of diveror plasmas evolving equations for all charge states of neon and deuterium; 2D transport simulations of tokamak equilibria in single-null X-point geometry with plasma ion and neutral atom species; and simulations of the time-dependent propagation of plasma filaments (blobs).Hermes-3 is publicly available on Github under the GPL-3 open source license. The repository includes documentation and a suite of unit, integrated and convergence tests. Program summaryProgram Title: Hermes-3CPC Library link to program files:https://doi.org/10.17632/rtkvc3r743.1Developer's repository link:https://github.com/bendudson/hermes-3Licensing provisions: GPLv3Programming language: C++14Nature of problem: Simulation of dynamics and steady-state solutions of multiple ion and neutral species in magnetised plasmas using drift-reduced fluid plasma models in 1 to 3 spatial dimensions. Focused on but not restricted to the simulation of turbulence in the boundary of tokamaks.Solution method: Hermes-3 implements a system of flexible software components, that are configured at run-time and coupled by passing a state object between them (the Encapsulate State and Command patterns). Hermes-3 builds on BOUT++ [1], employing the Method of Lines with implicit and explicit time-integration methods in curvilinear coordinates on block-structured meshes, making use of libraries including SUNDIALS, PETSc and Hypre. Hermes-3 implements drift-reduced plasma fluid equations using conservative finite difference methods, and atomic processes that couple plasma and neutral fluids.

New options for explicit all Mach number schemes by suitable choice of time integration methods

Many low-Mach or all-Mach number codes are based on space discretizations which in combination with the first order explicit Euler method as time integration would lead to an unstable scheme. In this paper, we investigate how the choice of a suitable explicit time integration method can stabilize these schemes. We restrict ourselves to some old prototypical examples in order to find directions for further research in this field.

A machine learning-based hybrid seismic analysis of a lead rubber bearing isolated building specimen

This paper presents the first experimental validation of the hybrid seismic analysis, previously developed by the authors, by simulating the structural behavior of a full-scale base-isolated building specimen tested at the E-Defense. The two-dimensional structural behavior of the lead rubber bearings (LRBs) is predicted by a machine learning model (MLM) previously developed by the authors, whereas analytical models describe the response of the superstructure. A Python script has been developed to incorporate both analytical models and MLMs into an explicit time integration method to perform three-dimensional hybrid seismic analyses. Moreover, conventional time history analyses have been carried out using a bilinear model for the isolation devices. Experimental, ML-based, and conventional analysis responses of 14 shake table motions are processed and compared in terms of hysteresis curves of the isolation layer, roof displacement and peak floor acceleration. Overall, hybrid seismic analysis showed better results than conventional one in terms of capturing the energy dissipation capacity of the isolation layer (maximum a20-index of 0.859 for the drift and 0.720 for the shear with respect to experiment). Similarly, roof displacement was predicted with a maximum a20-index value of 0.832. Besides, the maximum average computation time of hybrid seismic analysis was 1.78 s per integration time-step. Therefore, the efficiency of the hybrid seismic analysis is experimentally proven in this paper in terms of both accuracy and applicability. This machine learning-based analysis method is strongly recommended to simulate time-history response of complex structures.

A novel single-step explicit time integration method based on momentum corrector technique for structural dynamic analysis

In this study, a novel single-step explicit time integration method based on momentum corrector technique is proposed. The algorithmic procedure is formulated using cubic B-spline interpolation and Taylor series expansion. The momentum corrector is introduced to enhance solution accuracy of problems, especially for discontinuous load problems. Through the theoretical analysis of local truncation errors, spectral radius, algorithmic damping and period elongation, several parameter cases are obtained to achieve enhanced accuracy and stability. Numerical examples, including a standard single-degree-of-freedom problem, two discontinuous load problems, a seismic response analysis problem, a wave propagation problem and a nonlinear problem, are tested. The results illustrate the new method achieves high level of accuracy for discontinuous load problems, while simultaneously exhibiting significant accuracy improvement when dealing with other problems.

Multi-Level Solution Strategies for Implicit Time Discontinuous Galerkin Discretizations

The discontinuous Galerkin method and related flavors of high-order spectral element methods provide many well-known benefits for the spatial discretization of partial differential equations such as the Navier–Stokes equations. However, practical problems of engineering relevance such as large-eddy simulation of turbulent flows over complex geometries are computationally intractable by standard explicit time integration methods, necessitating the use of implicit methods. The efficient solution of the nonlinear algebraic systems arising from implicit time integration methods applied to DG discretizations of nonlinear PDEs is challenging; standard linearization methods result in very stiff block-sparse systems with prohibitive computational and memory requirements. This paper presents a low-memory, computationally efficient, implicit solution method that combines a framework of nonlinear polynomial multigrid, adaptive explicit Runge–Kutta smoothers, implicit Jacobian-free coarse level smoothers, nonlinear Krylov subspace acceleration and adaptive time stepping using feedback control of the nonlinear solver convergence rate.

Analytical and differential reformulations of the Vaiana–Rosati model for complex rate-independent mechanical hysteresis phenomena

A recent analytical model of hysteresis, developed by the authors to simulate a great variety of complex uniaxial rate-independent mechanical hysteresis phenomena, is reformulated in a twofold manner. First, it is proposed an analytical reformulation that, compared to the current version of the model, does not require the evaluation of any internal variable. Second, the closed form expressions provided by the analytical reformulation are expressed in rate form to foster its use especially in the field of nonlinear dynamics. To verify their accuracy, both formulations are first employed to reproduce four different complex hysteresis phenomena and the related results are compared in terms of generalized force–displacement hysteresis loops, tangent stiffness functions, and work-displacement relations. Subsequently, nonlinear time history analyses are carried out on four single degree of freedom mechanical systems for three different types of external generalized forces. To this end, two solution strategies are adopted. The former combines the model analytical reformulation with an explicit structure-dependent time integration method used to directly integrate the second-order ordinary differential equation of motion. The latter employs the model differential reformulation in conjunction with the Runge–Kutta method implemented to solve the equivalent system of three first-order ordinary differential equations. The comparison of the numerical results obtained by the two solution procedures shows an excellent agreement.

A novel three sub‐step explicit time integration method for wave propagation and dynamic problems

AbstractA novel explicit time integration method is formulated with sub‐step strategy and Cubic B‐spline interpolation method. Theoretical and numerical analysis are conducted to obtain optimized algorithm properties including algorithm accuracy, spectral stability and numerical dissipation/dispersion. A demonstrative dispersion analysis for wave propagation is presented to acquire optimal algorithm parameter value for finite element analysis of wave propagation problems. Numerical tests demonstrate that new method show desirable algorithm accuracy and convergence for dynamic problems. Especially, for highly nonlinear problems, the new method can provide very accurate and stable solutions.

Comparison of an Explicit and Implicit Time Integration Method on GPUs for Shallow Water Flows on Structured Grids

The accuracy, stability and computational efficiency of numerical methods on central processing units (CPUs) for the depth-averaged shallow water equations were well covered in the literature. A large number of these methods were already developed and compared. However, on graphics processing units (GPUs), such comparisons are relatively scarce. In this paper, we present the results of comparing two time-integration methods for the shallow water equations on structured grids. An explicit and a semi-implicit time integration method were considered. For the semi-implicit method, the performance of several iterative solvers was compared. The implementation of the semi-implicit method on a GPU in this study was a novel approach for the shallow water equations. This also holds for the repeated red black (RRB) solver that was found to be very efficient on a GPU. Additionally, the results of both methods were compared with several CPU-based software systems for the shallow water flows on structured grids. On a GPU, the simulations were 25 to 75 times faster than on a CPU. Theory predicts an explicit method to be best suited for a GPU due to the higher level of inherent parallelism. It was found that both the explicit and the semi-implicit methods ran efficiently on a GPU. For very shallow applications, the explicit method was preferred because the stability condition on the time step was not very restrictive. However, for deep water applications, we expect the semi-implicit method to be preferred.

An improved time-marching formulation based on an explicit time integration method for dynamic analysis of non-viscous damping systems

An improved time-marching formulation based on an explicit method [1] is proposed for non-viscous damping systems. In this method, an improved integral approximation scheme is developed to improve the convolution solution accuracy and is applicable to any causal kernel function. The mathematical derivation and calculation procedure based on the explicit method are formulated for non-viscous damping systems. For comparison, a composite explicit method proposed by Noh and Bathe is also extended for non-viscous damping systems. The proposed explicit method shows more desirable stability and accuracy properties than other competitive explicit methods. Numerical simulations of some representative examples demonstrate the proposed time-marching procedure is efficient for dynamic analysis of non-viscous damped systems.

Boundary and interface methods for energy stable finite difference discretizations of the dynamic beam equation

We consider energy stable summation by parts finite difference methods (SBP-FD) for the homogeneous and piecewise homogeneous dynamic beam equation (DBE). Previously the constant coefficient problem has been solved with SBP-FD together with penalty terms (SBP-SAT) to impose boundary conditions. In this work, we revisit this problem and compare SBP-SAT to the projection method (SBP-P). We also consider the DBE with discontinuous coefficients and present novel SBP-SAT, SBP-P, and hybrid SBP-SAT-P discretizations for imposing interface conditions. To demonstrate the methodology for two-dimensional problems, we also present a discretization of the piecewise homogeneous dynamic Kirchoff-Love plate equation based on the hybrid SBP-SAT-P method. Numerical experiments show that all methods considered are similar in terms of accuracy, but that SBP-P can be more computationally efficient (less restrictive time step requirement for explicit time integration methods) for both the constant and piecewise constant coefficient problems.

A novel sub-step explicit time integration method based on cubic B-spline interpolation for linear and nonlinear dynamics

A novel sub-step explicit time integration method is proposed. In the procedure, uniform cubic B-spline function interpolation is employed to achieve high-order accuracy. The proposed method has multiple free parameters which can be used to control algorithm properties. The proposed method shows at least second-order accuracy for linear systems with and without damping, and especially, when proper algorithmic parameters are selected, it can achieve third-order accuracy for no damping case. Numerical example analysis confirms higher accuracy than other classical explicit methods. For both linear and nonlinear dynamic problems, the proposed method shows desirable accuracy and efficiency.

Energy-conserving explicit and implicit time integration methods for the multi-dimensional Hermite-DG discretization of the Vlasov-Maxwell equations

We study the conservation properties of the Hermite-discontinuous Galerkin (Hermite-DG) approximation of the Vlasov-Maxwell equations. In this semi-discrete formulation, the total mass is preserved independently for every plasma species. Further, an energy invariant exists if central numerical fluxes are used in the DG approximation of Maxwell's equations, while a dissipative term is present when upwind fluxes are employed. In general, traditional temporal integrators might fail to preserve invariants associated with conservation laws during the time evolution. Hence, we analyze the capability of explicit and implicit Runge-Kutta (RK) temporal integrators to preserve such invariants. Since explicit RK methods can only ensure preservation of linear invariants but do not provide any control on the system energy, we consider modified explicit RK methods in the family of relaxation Runge-Kutta methods (RRK). These methods can be tuned to preserve the energy invariant at the continuous or semi-discrete level, a distinction that is important when upwind fluxes are used in the discretization of Maxwell's equations since upwind provides a numerical source of energy dissipation that is not present when central fluxes are used. We prove that the proposed methods are able to preserve the energy invariant and to maintain the semi-discrete energy dissipation (if present) according to the discretization of Maxwell's equations. An extensive set of numerical experiments corroborates the theoretical findings. It also suggests that maintaining the semi-discrete energy dissipation when upwind fluxes are used leads to an overall better accuracy of the method relative to using upwind fluxes while forcing exact energy conservation.

An improved explicit time-marching procedure with b-spline interpolation and weighted residual technique for structural seismic response analysis

In this work, an improved calculation procedure based on an explicit time integration method is formulated for structural seismic response analysis. The mathematical derivation and unified calculation procedure based on b-spline interpolation and weighted residual technique are given for seismic response analysis. The calculation procedure is improved for Bouc-Wen hysteresis nonlinear system. The explicit method has desirable algorithm properties including stability and accuracy properties. The stability interval of the proposed method is smaller than that of the composite method proposed by Noh and Bathe. The numerical examples illustrate the proposed method can achieve much high accuracy and efficiency compared with the competitive methods.

An asynchronous parallel explicit solver based on scaled boundary finite element method using octree meshes

Explicit time integration methods are an integral part of solving structural dynamics problems such as the propagation of elastic and acoustic waves, impact scenarios including crash tests and many more. One common limitation of this class of time integrators is, however, their conditional stability, meaning that highly distorted, small, or stiff finite elements typically govern the critical time step size. That is to say, even a small number of such elements will result in a significantly decrease of the feasible time step, which in turn drastically increases the computational costs of solving the semi-discrete equations of motion. This is especially true for non-uniform and unstructured meshes. Therefore, an asynchronous explicit solver in parallel is proposed in this article. The idea is to assign different time step sizes to different parts of the mesh, such that the overall computational effort is minimized. To improve the performance of the proposed solver, it is combined with a sophisticated octree meshing framework, where the balanced octrees are used, meaning that the ratio of the element sizes of adjacent elements cannot exceed a value of two. Thus, the critical time step of each element can be estimated straightforwardly. To avoid issues with hanging nodes, a special polyhedral element formulation, the scaled boundary finite element method, is employed. Since there are only a limited number of cell patterns in a balanced octree, the stiffness and mass matrices are pre-computed, significantly reducing the computational cost. Moreover, exploiting an element-by-element technique, the assembly of global stiffness matrices can be avoided. The main advantage of the described methodology is that the asynchronous explicit solver can be easily implemented in a high-performance computing environment. By means of several numerical examples, the accuracy and efficiency of the proposed method are demonstrated, as well as its versatility in handling complex engineering problems.