Nonlinear Structural Dynamics Problems Research Articles

A Robust and Efficient Composite Time Integration Algorithm for Nonlinear Structural Dynamic Analysis

This paper presents a new robust and efficient time integration algorithm suitable for various complex nonlinear structural dynamic finite element problems. Based on the idea of composition, the three-point backward difference formula and a generalized central difference formula are combined to constitute the implicit algorithm. Theoretical analysis indicates that the composite algorithm is a single-solver algorithm with satisfactory accuracy, unconditional stability, and second-order convergence rate. Moreover, without any additional parameters, the composite algorithm maintains a symmetric effective stiffness matrix and the computational cost is the same as that of the trapezoidal rule. And more merits of the proposed algorithm are revealed through several representative finite element examples by comparing with analytical solutions or solutions provided by other numerical techniques. Results show that not only the linear stiff problem but also the nonlinear problems involving nonlinearities of geometry, contact, and material can be solved efficiently and successfully by this composite algorithm. Thus the prospect of its implementation in existing finite element codes can be foreseen.

Dissipative, noniterative integration algorithms with unconditional stability for mildly nonlinear structural dynamic problems

A new family method is proposed to efficiently perform a nonlinear dynamic analysis with comparable accuracy. In fact, it integrates unconditional stability, explicit formulation and desired numerical damping together. Consequently, it is very promising for solving an inertia-type problem. The unconditional stability implies no limitation on step size, and an explicit formulation implies no iteration procedures. In addition, the desired numerical damping is continuously controlled, and it is possible to have zero damping. This damping can suppress the spurious high-frequency responses, while the low-frequency modes are almost unaffected and can be accurately integrated.

Multiscale methods for levitron problems: Theory and applications

A multiscale model based on magneto-static traps of neutral atoms or ion traps is described. The idea is to levitate a magnetic spinning top in the air, repelled by a base magnet. Real-life applications are related to magnetostatic trapping fields, e.g., [1], which allows trapping neutral atoms. In engineering, such effects are used in spectroscopy and atomic clocks, e.g., [2].Such problems are related to nonlinear problems in structural dynamics. The dynamics of such rigid bodies are modeled as a mechanical system with kinetic and potential parts, and can be described by a Hamiltonian, see [3–5].For such a problem, one must deal with different temporal and spatial scales, and so a novel splitting method for solving the levitron problem is proposed, see [6].In the present paper, we focus on explicit and extrapolated time-integrator methods, which are related to the Verlet algorithms. Due to the fact that we can decouple this multiscale problem into a kinetic part T and a potential part U, explicit methods are very appropriate. We try to limit the number of evaluations which are necessary (for a given accuracy) to obtain stable trajectories, and try to avoid the iterative cycles which are involved in implicit schemes, see [7].The kinetic and potential parts can be seen as generators of flows, see [5].The main problem is that of accurately formulating the Hamiltonian equation and this paper proposes a novel higher order splitting scheme to obtain stable states near the relative equilibrium. To improve the splitting scheme, a novel method, called MPE (multiproduct expansion method), is applied (see [8]), which includes higher order extrapolation schemes.The stability near this relative equilibrium is discussed with numerical studies using novel improved time-integrators. The best results are obtained with extrapolated Verlet schemes rather than higher order explicit Runge–Kutta schemes. Experiments are carried out with a magnetic top in an axisymmetric magnetic field (i.e., the levitron) and future applications to quantum computation will be discussed.

Insight into an implicit time integration scheme for structural dynamics

In Refs. [1,2], an effective implicit time integration scheme was proposed for the finite element solution of nonlinear problems in structural dynamics. Various important attributes were demonstrated. In particular, it was shown that the scheme remains stable, without the use of adjustable parameters, when the commonly used trapezoidal rule results in unstable solutions. In this paper we focus on additional important attributes of the scheme, and specifically on showing that the procedure can also be effective in linear analyses. We give, in comparison to other methods, the spectral radius, period elongation, and amplitude decay of the scheme and study the solution of a simple ‘model problem’ with a very flexible and stiff response.

Stability analysis of Rosenbrock-based Real-time compatible method via the energy approach

The focus of this paper is on stability analysis of 2-Stage Rosenbrock-based Real-time integrator for nonlinear structural dynamic problems, but the analysis spans both linear and nonlinear ranges. In linear case, the energy approach is presented to study its energy decaying property and high-frequency dissipation property. To investigate its nonlinear stability, numerical simulations are conducted on a shear-type structure including a pendulum. The theoretical analysis and the numerical simulations, on the one hand, efficiently demonstrate the applicability of the integrator to complicated structural problems; on the other hand, provide favorable approaches to investigate performances of other monolithic integrators.

New Practical Approach to Nonlinear Dynamic Analysis of Structures: Refinement of Newmark’s and Wilson’s Classical Methods

This paper presents a new method for direct time integration of nonlinear structural dynamic problems. In the proposed method, the order of time integration scheme is higher than the conventional Newmark’s family of methods. This method assumes second-order variation of the acceleration at each time step. Two variable parameters are used to increase the stability and accuracy of the method. The result obtained from this new higher-order method is compared with two implicit methods, namely the Wilson-θ and the Newmark’s average acceleration methods. The proposed method is an alternative quick method that can be written using matrix laboratory (MATLAB) for checking results from a more refined analysis or for a quick calculation with adequate accuracy.

Spectral analysis of a high-order Hermitian algorithm for structural dynamics

The spectral analysis of an efficient step-by-step direct integration algorithm for the structural dynamic equation is presented. The proposed algorithm is formulated in terms of two Hermitian finite difference operators of fifth-order local truncation error and it is unconditionally stable with no numerical damping presenting a fourth-order truncation error for period dispersion (global error). In addition, although it is in competition with higher-order algorithms presented in the literature, the computational effort is similar to that of the classical second-order Newmark’s method. The numerical application for nonlinear structural dynamic problems is also considered.

A harmonic balance approach for large-scale problems in nonlinear structural dynamics

Harmonic balance (HB) methods allow for rapid computation of time-periodic solutions for nonlinear dynamical systems. We present a filtered high dimensional harmonic balance (HDHB) approach, which operates in the time domain, and provide a framework for implementation into an existing finite element solver. To demonstrate its capabilities, the method is used to solve a set of nonlinear structural dynamics problems related to the field of flapping flight. For each example, the HDHB approach produces accurate steady-state solutions orders of magnitude faster than a traditional time-marching scheme.

Model reduction strategies for nonlinear beams subjected to large rotary actuations

AbstractThe solution to nonlinear structural dynamics problems with time marching schemes can be very expensive, particularly if the desired time-periodic response takes many cycles to form. Two cost reduction methods, which need not be considered separately, are formulated in this work. The first projects the nonlinear system of equations onto a reduced basis defined by a set of modes computed with proper orthogonal decomposition. The second utilises a monolithic time spectral element method, whereby the system of ordinary differential equations is converted into a single algebraic system of equations. The spectral element method can be formulated such that only the time-periodic response is computed. These techniques are implemented for a planar elastic beam, actuated at its base to emulate a flapping motion. Nonlinear elastic terms are computed with a corotational finite element method, while inertial terms are computed with a standard multibody dynamics formulation. For a variety of actuation frequencies and kinematic motions, results are given in terms of POD modes, reduced order model accuracy, and computational cost, for both the time marching and the monolithic time schemes.

Algorithms by Design: Part II—A Novel Normalized Time Weighted Residual Methodology and Design of a Family of Symplectic-Momentum Conserving Algorithms for Nonlinear Structural Dynamics

Previously, in Part I (see [1]) of this three-part exposition, we described for the first time for the particular class of LMS methods involving a single system and a single solve within each time step, which are the predominant methods in commercial and research software, how and why the equation of motion needs to be evaluated at precise time levels (which, in general, are not known to date) for ensuring correct algorithm designs. Here in Part II of this exposition, focusing attention on this class of LMS methods, a novel displacement based normalized time weighted residual approach is presented that naturally leads to new designs of a family of numerically non-dissipative (which are of focus here) symplectic-momentum conserving time integration operators without enforcing added or artificial constraints. The objectives are to foster the simulation of nonlinear structural dynamic problems for long-term durations. For illustration, the focus is on the Saint Venant Kirchoff material although the method can be readily extended to general material models (Part III ([2]) further shows extensions of this paper to additionally include controllable numerical dissipation for nonlinear dynamics applications to address issues associated with undesirable participation of the unresolved high frequency modes induced due to spatial discretization). The theoretical developments are described for nonlinear dynamics applications via extensions of the well-known Generalized Single Step Single Solve (GSSSS) linear dynamic framework, which encompasses the class of LMS methods in the single field form. In principle, this framework comprises two distinct classifications, namely, constrained U and V algorithmic architectures (characterized by the overshoot behavior) and was originally developed for designing linear dynamic algorithms using a generalization of the classical time weighted residual method. However, the problem with the classical approach is that it fails to adequately provide proper extensions to nonlinear dynamics applications as illustrated in this paper. In contrast to the classical approach, and also different from other approaches of designing algorithms, we show in this paper, via the new displacement based normalized time weighted residual approach, how to design symplectic-momentum time integrators that are implicit, and maintain second-order time accuracy for the class of general nonlinear inertial dynamics applications. For linear dynamic framework, the present concepts readily revert to the classical approach. Numerical illustrations are presented that demonstrate the computational and implementation aspects for nonlinear structural dynamics applications.

A time‐parallel implicit method for accelerating the solution of non‐linear structural dynamics problems

AbstractThe parallel implicit time‐integration algorithm (PITA) is among a very limited number of time‐integrators that have been successfully applied to the time‐parallel solution of linear second‐order hyperbolic problems such as those encountered in structural dynamics. Time‐parallelism can be of paramount importance to fast computations, for example, when space‐parallelism is unfeasible as in problems with a relatively small number of degrees of freedom in general, and reduced‐order model applications in particular, or when reaching the fastest possible CPU time is desired and requires the exploitation of both space‐ and time‐parallelisms. This paper extends the previously developed PITA to the non‐linear case. It also demonstrates its application to the reduction of the time‐to‐solution on a Linux cluster of sample non‐linear structural dynamics problems. Copyright © 2008 John Wiley & Sons, Ltd.

Domain-by-Domain Algorithm for Nonlinear Finite-Element Analysis of Structures

Parallel and distributed computers have been shown to provide the necessary computational power to solve large-scale engineering problems. However, in order for this computation style to be effectively used, efficient computational algorithms must be devised. In this work, a domain-by-domain algorithm is developed for the parallel solution of nonlinear structural dynamics problems. In the proposed algorithm, the original structure is partitioned into a number of subdomains. Each subdomain is solved independently and, therefore, concurrently using a traditional direct-solution method. Finally, the solution for the interface degrees of freedom between neighboring subdomains is obtained by enforcing compatibility and equilibrium using an iterative procedure. The nonlinear version of the algorithm involves two iterations: The nonlinear dynamic equilibrium iteration and the interface equilibrium and compatibility iteration. The integration of these two iterations is investigated and two strategies are developed. It is found that the strategy in which the two iterations are isolated is the most efficient. As a demonstration, the fully nonlinear transient analysis of a 20-story model building is carried out. Excellent accuracy in the solution and significant speed up values are obtained.

A multistep transversal linearization (MTL) method in non-linear dynamics through a Magnus characterization

A new form of a multi-step transversal linearization (MTL) method is developed and numerically explored in this study for a numeric-analytical integration of non-linear dynamical systems under deterministic excitations. As with other transversal linearization methods, the present version also requires that the linearized solution manifold transversally intersects the non-linear solution manifold at a chosen set of points or cross-section in the state space. However, a major point of departure of the present method is that it has the flexibility of treating non-linear damping and stiffness terms of the original system as damping and stiffness terms in the transversally linearized system, even though these linearized terms become explicit functions of time. From this perspective, the present development is closely related to the popular practice of tangent-space linearization adopted in finite element (FE) based solutions of non-linear problems in structural dynamics. The only difference is that the MTL method would require construction of transversal system matrices in lieu of the tangent system matrices needed within an FE framework. The resulting time-varying linearized system matrix is then treated as a Lie element using Magnus’ characterization [W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math., VII (1954) 649–673] and the associated fundamental solution matrix (FSM) is obtained through repeated Lie-bracket operations (or nested commutators). An advantage of this approach is that the underlying exponential transformation could preserve certain intrinsic structural properties of the solution of the non-linear problem. Yet another advantage of the transversal linearization lies in the non-unique representation of the linearized vector field – an aspect that has been specifically exploited in this study to enhance the spectral stability of the proposed family of methods and thus contain the temporal propagation of local errors. A simple analysis of the formal orders of accuracy is provided within a finite dimensional framework. Only a limited numerical exploration of the method is presently provided for a couple of popularly known non-linear oscillators, viz. a hardening Duffing oscillator, which has a non-linear stiffness term, and the van der Pol oscillator, which is self-excited and has a non-linear damping term.

Simulation of a Moving Elastic Beam Using Hamilton's Weak Principle

A proof-of-concept algorithm for calculating the general motion of elastic beams in the context of a multibody system analysis has been developed and validated. The governing equations for moving elastic beams with closed cross sections have been formulated using Hamilton’s law of varying action. They are then discretized with a mixed space–time finite element scheme, which results in a system of nonlinear algebraic equations. These equations are solved using unconstrained optimization techniques, thereby obtaining steady-state and time-accurate solutions for linear and nonlinear structural dynamics problems. Solutions obtained for a variety of static, steady-state, and transient test cases have demonstrated the accuracy of the formulation. Illustrative examples, including the cantilever elastica problem, rotation of a beam at a constant angular velocity, and prescribed flapping motion of a beam, are presented.

Error estimation for the HHT method in non-linear solid dynamics

This work presents a new error estimator that can be used in both linear and non-linear solid and structural dynamic problems. Specifically, this estimator is developed for the Hilber–Hughes–Taylor (HHT) method and it takes into account the error made in displacements and velocities. Since the accelerations computed by the HHT method are not sufficiently accurate to be used in the estimator, a strategy is presented to compute new accelerations with a minor computational cost. Finally, an efficient implementation of the estimator is designed.

A Trigonometric Recurrence Algorithm for Solving Nonlinear Problems in Structural Dynamics

An analytical-numerical methodology for solving nonlinear problems governed by partial differential equations (PDE) is presented. The authors have previously used a method named WEM that consists essentially of the statement of extended trigonometric series of uniform convergence (UC). Theorems, demonstrated previously, ensure the UC of the essential functions and the exactness of the eigenvalues. WEM has been applied to nonlinear dynamic problems in two different ways: As a direct variational method and as a solution in the classical sense. The present tool is applied in the second fashion and starts from the statement of the extended trigonometric series for all the unknown functions and derivatives involved in the PDE. The nonlinearities are treated in the same way. The application of consistence conditions leads to recurrence relationships among the coefficients of the extended series. For the sake of comparison an initial conditions problem (the well-known Duffing oscillator) is numerically solved. Then a beam example is solved in detail: A linear (both material and geometrical) supported beam, with end springs of nonlinear analytical response, under the action of a dynamic distributed load and/or prescribed initial conditions, is studied. Two numerical examples are included.

A novel non‐linearly explicit second‐order accurate L‐stable methodology for finite deformation: hypoelastic/hypoelasto‐plastic structural dynamics problems

AbstractA novel non‐linearly explicit second‐order accurate L‐stable computational methodology for integrating the non‐linear equations of motion without non‐linear iterations during each time step, and the underlying implementation procedure is described. Emphasis is placed on illustrative non‐linear structural dynamics problems employing both total/updated Lagrangian formulations to handle finite deformation hypoelasticity/hypoelasto‐plasticity models in conjunction with a new explicit exact integration procedure for a particular rate form constitutive equation. Illustrative numerical examples are shown to demonstrate the robustness of the overall developments for non‐linear structural dynamics applications. Copyright © 2004 John Wiley & Sons, Ltd.

A‐scalability and an integrated computational technology and framework for non‐linear structural dynamics. Part 1: Theoretical developments and parallel formulations

AbstractFor large‐scale problems and large processor counts, the accuracy and efficiency with reduced solution times and attaining optimal parallel scalability of the entire transient duration of the simulation for general non‐linear structural dynamics problems poses many computational challenges. For transient analysis, explicit time operators readily inherit algorithmic scalability and consequently enable parallel scalability. However, the key issues concerning parallel simulations via implicit time operators within the framework and encompassing the class of linear multistep methods include the totality of the following considerations to foster the proposed notion of A‐scalability: (a) selection of robust scalable optimal time discretized operators that foster stabilized non‐linear dynamic implicit computations both in terms of convergence and the number of non‐linear iterations for completion of large‐scale analysis of the highly non‐linear dynamic responses, (b) selecting an appropriate scalable spatial domain decomposition method for solving the resulting linearized system of equations during the implicit phase of the non‐linear computations, (c) scalable implementation models and solver technology for the interface and coarse problems for attaining parallel scalability of the computations, and (d) scalable parallel graph partitioning techniques. These latter issues related to parallel implicit formulations are of interest and focus in this paper. The former involving parallel explicit formulations are also a natural subset of the present framework and have been addressed previously in Reference 1 (Advances in Engineering Software 2000; 31: 639–647). In the present context, of the key issues, although a particular aspect or a solver as related to the spatial domain decomposition may be designed to be numerically scalable, the totality of the aforementioned issues simultaneously play an important and integral role to attain A‐scalability of the parallel formulations for the entire transient duration of the simulation and is desirable for transient problems. As such, the theoretical developments of the parallel formulations are first detailed in Part 1 of this paper, and the subsequent practical applications and performance results of general non‐linear structural dynamics problems are described in Part 2 of this paper to foster the proposed notion of A‐scalability. Copyright © 2003 John Wiley & Sons, Ltd.

Multidisciplinary Simulation of the Maneuvering of an Aircraft

A computational methodology for the simulation of the transient aeroelastic response of an unrestrained and flexible aircraft during high-G maneuvers is presented. The key components of this methodology are: (a) a three-field formulation for coupled fluid/structure interaction problems; (b) a second-order time-accurate and geometrically conserva- tive flow solver for CFD computations on unstructured dynamic meshes; (c) a corotational finite element method for the solution of geometrically nonlinear and unrestrained structural dynamics problems; (d) a robust method for updat- ing an unrestrained and unstructured moving fluid mesh; and (e) a second-order time-accurate staggered algorithm for time-integrating the coupled fluid/structure semi-discrete equations of motion. This computational methodology is illus- trated with the simulation on a parallel processor of several three-dimensional high-G pullup maneuvers of the Langley Fighter in the transonic regime, using a detailed finite element aeroelastic model.

Adaptive finite element procedures for linear and non-linear dynamics

This paper discusses implementation and adaptivity of the Discontinuous Galerkin (DG) finite element method as applied to linear and non-linear structural dynamic problems. By the DG method, both displacements and velocities are approximated as piecewise bilinear functions in space and time and may be discontinuous at the discrete time levels. Both implicit and explicit iterative algorithms for solving the resulted system of coupled equations are derived. They are third-order accurate and, while the implicit procedure is unconditionally stable, the explicit one is conditionally stable. An h-adaptive procedure based on the Zienkiewicz–Zhu error estimate using the SPR technique is applied. Numerical examples are presented to show the suitability of the DG method for both linear and non-linear structural dynamic analysis. Copyright © 1999 John Wiley & Sons, Ltd.