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Hybrid methods for solving structural geometric nonlinear dynamic equations: Implementation of fifth-order iterative procedures within composite time integration methods

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RESULTS IN ENGINEERING 2024年 21卷

作者： Shahraki, Mojtaba Shahabian, Farzad Maghami, Ali Ferdowsi Univ Mashhad Civil Engn Dept Azadi Sq Mashhad *** Razavi Khorasan Iran Politecn Bari Dept Mech Math & Management Via Orabona 4 I-70125 Bari Italy

This paper studies a class of hybrid methods that implement multi-point iterative procedures as the nonlinear solver within optimized composite implicit methods. These multi-point methods are employed to enhance convergence order and accuracy without requiring higher-order derivatives for solving structural geometric nonlinear dynamic equations. The promising approach provides an opportunity to delve deeper into its implications and explore the potential applications of multi-point techniques and composite methods across the analysis of dynamical systems. This study involves the utilization of two-point and five-point iterative methods with fifth-order convergence within three-step and four-step optimized composite implicit time integration methods. Moreover, a comprehensive hybrid scheme is introduced by integrating a family of optimized composite implicit methods with single-point and multi-point iterative methods. Then, the performance of the hybrid scheme is analyzed through structural examples, considering factors such as time step size, tolerance threshold, and the degrees of freedom of structures. The findings and numerical results illustrate that multi-point methods outperform the single-point method in terms of the number of iterations. Simultaneously, the four-step time integration method exhibits superior performance compared to the three-step time integration method, albeit with a marginal difference. It is observed that the hybrid method, consisting of the four-step time integration method and the five-point iterative method, performs better than other methods in terms of the number of iterations and errors for solving geometric nonlinear dynamic equations, especially in large structures. Thus, it stands out as a favorable choice for practical analyses and a possible candidate for generalization to other types of nonlinear problems in computational mechanics.

关键词： Dynamic analysis Geometric nonlinear time integration methods Implicit methods Multi-point methods

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Performance of implicit A-stable time integration methods for multibody system dynamics

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MULTIBODY SYSTEM DYNAMICS 2022年第3期54卷 263-301页

作者： Zhang, Huimin Zhang, Runsen Zanoni, Andrea Masarati, Pierangelo Beihang Univ Sch Aeronaut Sci & Engn Beijing 100083 Peoples R China Politecn Milan Dipartimento Sci & Tecnol Aerosp I-20156 Milan Italy

This paper illustrates the performance of several representative implicit A-stable time integration methods with algorithmic dissipation for multibody system dynamics, formulated as a set of mixed implicit first-order differential and algebraic equations. The integrators include the linear multi-step methods with two to four steps, the single-step reformulations of the linear multi-step methods, and explicit first-stage, singly diagonally-implicit Runge-Kutta methods. All methods are implemented in the free, general-purpose multibody solver MBDyn. Their formulations and implementation are presented. According to the comparison from linear analysis and numerical experiments, some general conclusions on the selection of integration schemes and their implementation are obtained. Although all of these methods can predict reasonably accurate solutions, the specific advantages that each of them has in different situations are discussed.

关键词： Implicit A-stability time integration methods Multibody system dynamics

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Comparison between recent implicit time integration methods with frequency dissipation for nonlinear structural applications

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LATIN AMERICAN JOURNAL OF SOLIDS AND STRUCTURES 2022年第3期19卷

作者： Fernandes, William Luiz Barbosa, Gustavo Botelho Greco, Marcelo da Mota Silveira, Ricardo Azoubel Pontificia Univ Catolica Minas Gerais Dept Engn Civil R Dom Jose Gaspar 500 BR-30535901 Belo Horizonte MG Brazil Univ Fed Minas Gerais Dept Engn Estruturas Av Pres Antonio Carlos 6627 BR-31270901 Belo Horizonte MG Brazil Univ Fed Minas Gerais Dept Engn Estruturas Av Pres Antonio Carlos 6627 BR-31270901 Belo Horizonte MG Brazil Univ Fed Ouro Preto Escola Minas Dept Engn Civil Campus Univ S-N BR-35400000 Ouro Preto MG Brazil

The present paper aims to test recent (Truly self-starting two sub-step method and three-parameter single-step implicit method) and classical (Generalized-alpha, HHT-alpha, and WBZ-alpha methods) time integration methods using the geometrically nonlinear Positional Finite Element Method (PFEM). The numerical formulation is based on the total Lagrangian approach and uses the Hessian matrix to obtain the response. The mixed hardening inelastic model applied to PFEM is also presented. Two examples validate the time integration algorithms and the inelastic model. In the first example, the mixed hardening inelastic model is compared with the the bilinear stress-strain model and the elastic-perfectly plastic hinge model, and aspects such as amplitude decay and period elongation are discussed. In the second example, the implemented algorithms are verified in a severe geometrically nonlinear example, considering the influence of numerical dissipation, time interval, and the number of elements in the response. Results show the relevance of numerical damping for numerical stabilization and the good performance of the Generalized-alpha algorithm.

关键词： Geometrical Nonlinearity Mixed Hardening Model time integration methods Positional Finite Element Formulation Numerical Dissipation Control Structural Behavior

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Three time integration methods for incompressible flows with discontinuous Galerkin Boltzmann method

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2018年第11期75卷 4091-4106页

作者： Shao, Weidong Li, Jun Xi An Jiao Tong Univ Inst Turbomachinery Xian 710049 Shaanxi Peoples R China Collaborat Innovat Ctr Adv Aeroengine Beijing 100191 Peoples R China

This paper presents three time integration methods for incompressible flows with finite element method in solving the lattice-BGK Boltzmann equation. The space discretization is performed using nodal discontinuous Galerkin method, which employs unstructured meshes with triangular elements and high order approximation degrees. The time discretization is performed using three different kinds of time integration methods, namely, direct, decoupling and splitting. From the storage cost, temporal accuracy, numerical stability and time consumption, we systematically compare three time integration methods. Then benchmark fluid flow simulations are performed to highlight efficient time integration methods. Numerical results are in good agreement with others or exact solutions. (C) 2018 Elsevier Ltd. All rights reserved.

关键词： time integration methods Nodal discontinuous Galerkin method Lattice-BGK Boltzmann equation Incompressible flows Stiff terms

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Investigation of the time integration methods on the Parareal Method for Field Computation of Eddy Currents Problems 17

Investigation of the Time Integration Methods on the Pararea...

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17th IEEE Biennial Conference on Electromagnetic Field Computation (IEEE CEFC)

作者： Niyonzima, I. Clemens, M. Schoeps, S. Tech Univ Darmstadt Inst Theorie Elektromagnetischer Felder Darmstadt Germany Tech Univ Darmstadt Grad Sch Computat Engn Darmstadt Germany Univ Wuppertal Chair Electromagnet Theory Wuppertal Germany

ISBN: (纸本)9781509010325

In this paper, we investigate the influence of time integration methods on the performance of the Parareal method for the computation of eddy current problems. Parareal is a method that allows parallelization in time domain. The time interval is split into many smaller time intervals (as many as the number of available CPUs) with a fine time grid and solved in parallel allowing to capture the finescale details of the solution. An approximation of initial conditions for these fine problems is obtained by solving a cheap, time-dependent problem defined on a coarse grid for the entire time interval. The method has been successfully implemented for the problem of eddy currents using the implicit backward Euler method. In this paper we will investigate the influence of the time stepping methods on the convergence and the complexity of the Parareal algorithm.

关键词： Eddy currents Parallel-in-time methods Parareal time integration methods

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OXFORD-UMAT: An efficient and versatile crystal plasticity framework

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INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES 2025年 307卷

作者： Demir, Eralp Martinez-Pechero, Alvaro Hardie, Chris Tarleton, Edmund Univ Oxford Dept Engn Sci Parks Rd Oxford OX1 3PJ England Culham Sci Ctr Atom Energy Author Abingdon OX14 3DB England

The crystal plasticity-based finite element method is widely used, as it allows complex microstructures to be simulated and allows direct comparison with experiments. This paper presents the OXFORD-UMAT for Abaqus (R), a novel crystal plasticity code that is publicly available online for researchers interested in using crystal plasticity. The model is able to simulate a wide range of materials and incorporates two different solvers based on the solution of slip increments and Cauchy stress, with variants of state update procedures including explicit, semi-implicit, and fully-implicit for computational efficiency that can beset by the user. Constitutive laws are available fora range of materials with single or multiple phases for slip, creep, strain hardening, and back stress. The model includes geometrically necessary dislocations that can be computed using finite element interpolation functions by four alternative methods, including the total form with and without a correction for the dislocation flux, a widely used rate form, and a slip-gradient formulation. In addition, the initial strengthening and subsequent softening seen in irradiated materials can also be simulated with the model. The analysis is available in 2D (plane stress and plane strain) and 3D, including linear and quadratic elements. Here we include full derivations of the key equations used in the code and then demonstrate the capability of the code by modeling single-crystal and large-scale polycrystal cases. Comparison of OXFORDUMAT with other available crystal plasticity codes for Abaqus (R) reveals the efficiency of the proposed approach, with the backup solver offering greater versatility for handling convergence issues commonly found in practical applications.

关键词： Crystal plasticity Finite element method Abaqus UMAT time integration methods update Forward-gradient method

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A single-step recursive algorithm of the convolution integral for computing non-viscous damping forces in dynamic analyses

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MECHANICAL SYSTEMS AND SIGNAL PROCESSING 2024年 208卷

作者： Du, Xian Ting Xia, Chaoyi Meng, Chunyan Beijing Jiaotong Univ Sch Civil Engn Beijing 100044 Peoples R China

The non-viscous damping force features the convolution integral of the velocity and the kernel function, causing the difficulty of the solutions of the system equation of motion due to the nonlocal time dependence. In accordance with the linear time-invariant system theory in signal processing, this paper presents a single-step recursive algorithm of convolution integral for efficiently computing non-viscous damping forces. The algorithm parameters are effectively identified from the discrete responses of the kernel function to the rectangular unit impulse through digital filter technologies. The proposed algorithm is compatible with the solution schemes based on existing time integration methods, which handle the nonlocal dependence using internal recursive support vectors while leaving the order of the equation of motion unchanged. The central difference method, the Newmark-ss method, and the Bathe method are employed to illustrate the use of the algorithm, proving to retain the stability characteristics of time integration methods. The single and multiple DOF examples show that the simulated results based on the compatible solution schemes compare well with those built on other existing methods, verifying high accuracy and efficiency. The proposed algorithm thus extends the application range of existing time integration methods from the viscously damped system to the non-viscously damped one, fully demonstrating its generalizability.

关键词： Non-viscous damping Convolution integral Non -local time dependence The linear time-invariant system theory time integration methods

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Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear Schrodinger equations

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IMA JOURNAL OF NUMERICAL ANALYSIS 2021年第1期41卷 594-617页

作者： Blanes, Sergio Casas, Fernando Gonzalez, Cesareo Thalhammer, Mechthild Univ Politecn Valencia Inst Matemat Multidisciplinar E-46022 Valencia Spain Univ Jaume 1 Dept Matemat IMAC Castellon de La Plana 12071 Spain Univ Valladolid Dept Matemat Aplicada Valladolid 47011 Spain Leopold Franzens Univ Innsbruck Inst Math A-6020 Innsbruck Austria

This work is devoted to the derivation of a convergence result for high-order commutator-free quasi-Magnus (CFQM) exponential integrators applied to nonautonomous linear Schrodinger equations;a detailed stability and local error analysis is provided for the relevant special case where the Hamilton operator comprises the Laplacian and a regular space-time-dependent potential. In the context of nonautonomous linear ordinary differential equations, CFQM exponential integrators are composed of exponentials involving linear combinations of certain values of the associated time-dependent matrix;this approach extends to nonautonomous linear evolution equations given by unbounded operators. An inherent advantage of CFQM exponential integrators over other time integration methods such as Runge-Kutta methods or Magnus integrators is that structural properties of the underlying operator family are well preserved;this characteristic is confirmed by a theoretical analysis ensuring unconditional stability in the underlying Hilbert space and the full order of convergence under low regularity requirements on the initial state. Due to the fact that convenient tools for products of matrix exponentials such as the Baker-Campbell-Hausdorff formula involve infinite series and thus cannot be applied in connection with unbounded operators, a certain complexity in the investigation of higher-order CFQM exponential integrators for Schrodinger equations is related to an appropriate treatment of compositions of evolution operators;an effective concept for the derivation of a local error expansion relies on suitable linearisations of the evolution equations for the exact and numerical solutions, representations by the variation-ofconstants formula and Taylor series expansions of parts of the integrands, where the arising iterated commutators determine the regularity requirements on the problem data.

关键词： nonautonomous linear evolution equations Schrodinger equations quantum systems time integration methods exponential integrators Magnus integrators commutator-free quasi-Magnus exponential integrators stability local error convergence

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An enhanced explicit time-marching technique for wave propagation analysis considering adaptive time integrators

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COMPUTER methods IN APPLIED MECHANICS AND ENGINEERING 2020年 363卷

作者： Soares, Delfim, Jr. Univ Fed Juiz de Fora Struct Engn Dept BR-36036330 Juiz De Fora MG Brazil

In this work, a novel family of explicit time marching techniques is proposed for wave propagation analyses. The main features of the new approach are: (i) the family is based only on single-step displacement-velocity relations, being very simple to implement;(ii) as usual in explicit analyses, it requires no system of equations to be dealt with, being very efficient;(iii) it is truly self-starting;(iv) it is second-order accurate;(v) it enables advanced controllable algorithmic dissipation in the higher modes;(vi) it is locally defined and based on adaptive time integrators, allowing enhanced accuracy to be obtained;(vii) it is function of the adopted temporal and spatial discretizations, as well as of the properties of the model and computed responses, establishing a link between these features;(viii) it is entirely automated, requiring no expertise from the user;(ix) it stands as an adaptive non-iterative approach, further improving its efficiency;(x) it is conditionally stable and its stability limit is twice that of the Central Difference Method;(xi) the stability limit of the method remains unaltered, despite the introduction of dissipation. The proposed technique is very versatile and effective, providing the main positive attributes that are requested from an efficacious explicit time marching procedure. Numerical results are presented at the end of the manuscript, illustrating the excellent performance of the novel methodology. (C) 2020 Elsevier B.V. All rights reserved.

关键词： time integration methods Explicit analysis Hyperbolic models Adaptive time integrators Stability Dissipation control

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A model/solution-adaptive explicit-implicit time-marching technique for wave propagation analysis

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INTERNATIONAL JOURNAL FOR NUMERICAL methods IN ENGINEERING 2019年第7期119卷 590-617页

作者： Soares, Delfim, Jr. Univ Fed Juiz de Fora Struct Engn Dept BR-36036330 Juiz De Fora MG Brazil

In this work, an explicit-implicit time-marching procedure with model/ solution-adaptive time integration parameters is proposed for the analysis of hyperbolic models. The two time integrators of the methodology are locally evaluated, enabling their different spatial and temporal distributions. The first parameter defines the explicit/implicit subdomains of the model, and it is defined in a way that stability is always ensured, as well as period elongation errors are reduced;the second parameter controls the dissipative properties of the methodology, allowing spurious high-frequency modes to be properly eliminated, rendering reduced amplitude decay errors. In addition, the proposed explicit-implicit approach allows contracted systems of equations to be obtained, reducing the computational effort of the analysis. The main features of the novel methodology can be summarized as follows: (i) it is simple;(ii) it is locally defined;(iii) it has guaranteed stability;(iv) it is an efficient noniterative single-step procedure;(v) it provides enhanced accuracy;(vi) it enables advanced controllable algorithmic dissipation in the higher modes;(vii) it considers a link between the temporal and the spatial discretization;(viii) it stands as a single-solve framework based on reduced systems of equations;(ix) it is truly self-starting;and (x) it is entirely automatic.

关键词： accuracy adaptive parameters explicit implicit analysis hyperbolic models stability time integration methods

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