Abstract: Usually the notion “Newton-Raphson method” is used in the context of non-linear finite element analysis based on quasi-static problems in solid mechanics. It is pointed out that this is only true in the case of non-linear elasticity. In the case of constitutive equations of evolutionary-type, like in viscoelasticity, viscoplasticity or elastoplasticity, the “Multilevel-Newton algorithm” is usually applied yielding the notions of global and local level (iteration), as well as the consistent tangent operator. In this paper, we investigate the effects of a consistent application of the classical Newton-Raphson method in connection with the finite element method, and compare it with the classical Multilevel-Newton algorithm. Furthermore, an improved version of the Multilevel-Newton method is applied.

Abstract: The Newmark method for the numerical integration of second order equations has been extensively used and studied along the past fifty years for structural dynamics and various fields of mechanical engineering. Easy implementation and nice properties of this method and its derivatives for linear problems are appreciated but the main drawback is the treatment of discontinuities. Zienkiewicz proposed an approach using finite element concept in time, which allows a new look at the Newmark method. The idea of this paper is to propose, thanks to this approach, the use of a time partition of the unity method denoted Time Extended Finite Element Method (TX-FEM) for improved numerical simulations of time discontinuities. An enriched basis of shape functions in time is used to capture with a good accuracy the non-polynomial part of the solution. This formulation allows a suitable form of the time-stepping formulae to study stability and energy conservation. The case of an enrichment with the Heaviside function is developed and can be seen as an alternative approach to time discontinuous Galerkin method (T-DGM), stability and accuracy properties of which can be derived from those of the TX-FEM. Then Space and Time X-FEM (STX-FEM) are combined to obtain a unified space–time discretization. This combined STX-FEM appears to be a suitable technique for space–time discontinuous problems like dynamic crack propagation or other applications involving moving discontinuities. Copyright © 2005 John Wiley & Sons, Ltd.

Abstract: This paper deals with an explicit numerical integration method for real-time pseudo dynamic tests. The proposed method, termed the MPC-SSP method, is suited to use in real-time pseudo dynamic tests as no iteration steps are involved in each step of computation. A procedure for implementing the proposed method in real-time pseudo dynamic tests is described in the paper. A state-space approach is employed in this study to formulate the equations of motion of the system, which is advantageous in real-time pseudo dynamic testing of structures with active control devices since most structural control problems are formulated in state space. A stability and accuracy analysis of the proposed method was performed based on linear elastic systems. Owing to an extrapolation scheme employed to predict the system's future response, the MPC-SSP method is conditionally stable. To demonstrate the effectiveness of the MPC-SSP method, a series of numerical simulations were performed and the performance of the MPC-SSP method was compared with other pseudo dynamic testing methods including Explicit Newmark, Central Difference, Operator Splitting, and OS-SSP methods based on both linear and non-linear single-degree-of-freedom systems. Copyright © 2004 John Wiley & Sons, Ltd.

Abstract: The bi-potential method has been successfully applied to the modeling of frictional contact problems in static cases. This paper presents an extension of this method for dynamic analysis of impact problems with deformable bodies. A first order algorithm is applied to the numerical integration of the time-discretized equation of motion. Using the Object-Oriented Programming (OOP) techniques in C++ and OpenGL graphical support, a finite element code including pre/postprocessor FER/Impact is developed. The numerical results show that, at the present stage of development, this approach is robust and efficient in terms of numerical stability and precision compared with the penalty method.

Abstract: An adaptive finite element procedure is developed for modelling transient phenomena in elastic solids, including both wave propagation and structural dynamics. Although both temporal and spatial adaptivity are addressed, the novel feature of the formulation is the use of mesh superposition to produce spatial refinement (referred to as s-adaptivity) in transient problems. Spatial error estimation is based on superconvergent patch recovery of higher-order accurate stresses and is used to guide mesh adaptivity, while the temporal error estimation is based on the assumption of linearly varying third-order time derivatives of the displacement field and is used to adjust the time step size for the HHT-α variant of the Newmark direct numerical integration method. Spatial adaptivity of the mesh is performed using a hierarchical h-refinement scheme that is efficiently implemented using a structured version of finite element mesh superposition. This particular spatial adaptivity scheme is extremely fast and consequently makes it feasible to repeatedly update both the mesh and the time increment as required in an adaptive transient analysis. This work represents the initial effort in applying this type of spatial adaptivity to transient problems. Three example problems are given to demonstrate the performance characteristics of the s-adaptive procedure. Copyright © 2005 John Wiley & Sons, Ltd.

Abstract: The Newmark formulation for the time-stepping solution of the electric field vector wave equation is modified in order to suppress low-frequency spurious responses. It is verified that the proposed method has the advantage of halving the number of iterations necessary to solve the system of linear equations and it is shown that setting the number of iterations per time step can be more convenient than establishing a residual tolerance to verify convergence. In addition, the use of diagonal and incomplete Cholesky preconditioners are tested through numerical examples and they appear to be equivalent in terms of CPU time, leaving the first as the preferable choice concerning memory storage and programing facilities.

Abstract: When simulating the behavior of a mechanical system, the time evolution of the generalized coordinates used to represent the configuration of the model is computed as the solution of a combined set of ordinary differential and algebraic equations (DAEs). There are several ways in which the numerical solution of the resulting index 3 DAE problem can be approached. The most well-known and time-honored algorithms are the direct discretization approach, and the state-space reduction approach, respectively. In the latter, the problem is reduced to a minimal set of potentially new generalized coordinates in which the problem assumes the form of a pure second order set of Ordinary Differential Equations (ODE). This approach is very accurate, but computationally intensive, especially when dealing with large mechanical systems that contain flexible parts, stiff components, and contact/impact. The direct discretization approach is less but nevertheless sufficiently accurate yet significantly faster, and it is the approach that is considered in this paper. In the context of direct discretization methods, approaches based on the Backward Differentiation Formulas (BDF) have been the traditional choice for more than 20 years. This paper proposes a new approach in which BDF methods are replaced by the Newmark formulas. Local convergence analysis is carried out for the proposed method, and step-size control, error estimation, and nonlinear system solution related issues are discussed in detail. A series of two simple models are used to validate the method. The global convergence analysis and a computational-efficiency comparison with the most widely used numerical integrator available in the MSC.ADAMS commercial simulation package are forthcoming. The new method has been implemented successfully for industrial strength Dynamic Analysis simulations in the 2005 version of the MSC.ADAMS software and used very effectively for the simulation of systems with more than 15,000 differential-algebraic equations.Copyright © 2005 by ASME

Abstract: NUMERICAL MODEL GENERATION Finite Element Analysis Solution of Boundary Value Problems Finite Element Shape Functions Finite Element Basis Functions Assembly of Finite Element Matrices Element Matrix Generation Local to Global Coordinate Transformation A Quadrilateral Finite Element References Finite Element Model Generation Spline Approximation Geometric Modeling Objects Geometric Model Discretization Delaunay Mesh Generation References Modeling of Physical Phenomena Lagrange's Equations of Motion Continuum Mechanical Systems Finite Element Analysis of Elastic Continuum A Tetrahedral Finite Element Equation of Motion of Mechanical System Transformation to Frequency Domain References Constraints and Boundary Conditions The Concept of Multi-Point Constraints The Elimination of Multi-Point Constraints The Axial Bar Element The Concept of Single Point Constraints The Elimination of Single Point Constraints References Singularity Detection of Finite Element Models Local Singularities Global Singularities Massless Degrees of Freedom Industrial Case Studies References COMPUTATIONAL REDUCTION TECHNIQUES Matrix Factorization and Linear System Solution Finite Element Matrix Reordering Sparse Matrix Factorization Multifrontal Factorization Linear System Solution Distributed Factorization and Solution Factorization Case Study References Static Condensation Single Level, Single Component Condensation Computational Example Single Level, Multiple Component Condensation Multiple Level Static Condensation Static Condensation Case Study References Spectral Computations Spectral Transformation Lanczos Reduction Generalized Eigenvalue Problem Eigenvalue Computation Distributed Eigenvalue Computation Normal Modes Analysis Case Study Complex Spectral Computations Complex Modes Analysis Case Study Dense Eigenvalue Analysis Householder Reduction Techniques Tridiagonal Reduction Reduction to Hessenberg Form References Dynamic Reduction Single Level, Single Component Dynamic Reduction Accuracy of Dynamic Reduction Computational Example Single Level, Multiple Component Dynamic Reduction Multiple Level Dynamic Reduction Multibody Analysis Application References Component Modal Synthesis Single Level, Single Component Modal Synthesis Mixed Boundary Component Mode Reduction Computational Example Single Level, Multiple Component Modal Synthesis Multiple Level Modal Synthesis Component Modal Synthesis Case Study References ENGINEERING SOLUTION COMPUTATIONS Modal Solution Technique Modal Reduction Truncation Error in Modal Reduction The Method of Residual Flexibility The Method of Mode Acceleration Coupled Modal Solution Application References Transient Response Analysis The Central Difference Method The Newmark Method Starting Conditions and Time Step Changes Stability of Time Integration Techniques Transient Solution Case Study References Frequency Domain Analysis Direct Frequency Response Analysis Reduced Order Frequency Response Analysis Accuracy of Reduced Order Solution Frequency Response Case Study References Nonlinear Analysis Introduction to Nonlinear Analysis Newton-Raphson Methods Quasi-Newton Iteration Techniques Convergence Criteria Computational Example Nonlinear Dynamics References Sensitivity and Optimization Design Sensitivity Design Optimization Planar Bending of the Bar Computational Example Eigenfunction Sensitivities Variational Analysis References Engineering Result Computations Displacement Recovery Stress Calculation Nodal Data Interpolation Level Curve Computation Engineering Results Case Study References Closing Remarks Annotation Index

Abstract: Four kinds of moving mass elements, 1st-node, 2nd-node, full and short-range mass elements, are presented, where the 1st-node (or 2nd-node) mass element refers to that with mass distributed from the first node (or second node) to the arbitrary position of a two-node beam element, the full mass element is the special case of the 1st-node (or 2nd-node) mass element with mass distributed over the full length of the beam element, while the short-range mass element is the case with its location arbitrary on a beam element. If the total range of a distributed mass is denoted by R and the length of each beam element is denoted by , then, for the case of R≥, one may model the distributed mass on the beam using the combination of the 1st-node, 2nd-node and full mass elements, while for the case of R <, one may model the distributed mass using the short-range mass element. It has been found that the effects of the vertical (y) and horizontal (x) inertia forces, Coriolis force and centrifugal force induced by the moving distributed mass can be easily taken into the formulations by means of the last concept. To illustrate the application of the presented theory, the dynamic analysis of a pinned-pinned beam and that of a portal frame under the action of a moving uniformly distributed mass are performed by means of the finite element method and the Newmark integration method. Numerical results show that some pertinent factors, such as Coriolis force, centrifugal force, acceleration, velocity and total range of the moving distributed mass, have significant influences on the vertical (y) and horizontal (x) response of a structure.

Abstract: Introduction of algorithmic damping by increasing the parameter values in the Newmark algorithm leads to undesirable low-frequency damping and reduced order of accuracy It is demonstrated, how these effects can be removed by introducing an extra damping term in the form of a first order linear filter When the linear filter is discretized in time, the state variable associated with the filter can be eliminated, leading to a weighted average of the equations of motion at two consecutive times The filter procedure contains the known versions of alpha weighted Newmark methods as special cases, but gives a different and improved weighting of the excitation terms A complete analysis of the properties of the algorithm when used on linear systems is given, including the frequency response properties It is demonstrated that the effect of ‘overshoot’ is the consequence of a conservation relation that operates on a modified form of the mechanical energy of the system, and analytic results are presented for the magnitude of the effect Copyright © 2005 John Wiley & Sons, Ltd

Abstract: Direct time integration is used widely for the solution of large deformation problems in solid and structural mechanics. Direct integration schemes that are unconditionally stable for linear dynamic problems are also used for nonlinear problems. However, unconditional stability may be lost in the nonlinear regime. The Newmark method (trapezoidal rule) is used widely but may become unstable when large deformations and long time durations are considered. A composite scheme is proposed for such analysis cases, and the results obtained using the trapezoidal rule and the composite formula in a test problem are given. These results indicate the value of the composite scheme.

Abstract: We propose a numerical method for dealing with interactions between multiple particles and complex structures. In the method, the structures are represented on a grid by using the level set method. The interactions of particles and structures are calculated by a method based on the discrete element method. The method can treat the interaction between multiparticle and complex structures robustly.

Abstract: In this note we illustrate how to obtain the full family of Newmark 's time integration algorithms within a rigorous variational framework, i.e., by discretizing suitably defined extended functionals, rather than by starting from a weak form (for instance, of the Galerkin type), as done in the past. The availability of functionals as a starting point is useful both as a tool to obtain new families of time integration methods, and as a theoretical basis for error estimates. To illustrate the first issue, here we provide some examples of how to obtain modified algorithms, in some cases significantly more accurate than the basic Newmark one despite having a comparable computational cost.

Abstract: A planetary gear is modeled in this work. The equations of motion and the eigenfrequencies are recovered. The computation of the dynamic response is made using the spectral iterative method. The procedure is based on a modal approach with developments in the frequency domain. This technique has been successfully used and its convergence was quickly reached. Good agreement was obtained compared with the standard Newmark method. The dynamic response of the planetary gear is given directly in the frequency domain. The inverse Fourier transform gives the time response of the system.

Abstract: As to the characteristic of large wind-induced response, an aeroelastic model of single-rod transmission tower was tested in a wind tunnel to study its response to different wind velocity. Under different circumstances with and without wires, the dynamic along-wind response and across-wind response in the turbulence flow and the aerodynamic force coefficients in the laminar flow were measured respectively. on the basis of that, the maximum acceleration, displacement, the proportion of across-wind response to (along-wind) response and the damping effect of wires were obtained, all of which can be used in structure (design.) Furthermore, the dynamic response was calculated with triangle series superposition method and Newmark method in time domain and with the power spectrum integral method in frequency domain respectively. The results were compared and the accuracy of the test was verified.

Abstract: At present, most researches on the vortex-induced vibration of submarine free spanning pipelines ignore the effect of internal flowing fluid; furthermore, there are no research reports considering the coupling effect of internal and external fluid with the free span. In this paper, combining Iwan's wake oscillator model with the differential equation derived for the dynamic response of submarine free spanning pipelines with inclusion of internal flow, the pipe-fluid coupling equations are developed to investigate the effect of internal flow on the vortex-induced vibration of the free spans. The finite element approximation is implemented to derive the matrix equations of equilibrium. The Newmark method combined with simple iteration is used to solve the system of equations. The results indicate that the internal fluid flow may cause the shift of resonance band to the lower frequency and a slight decrease in the peak value; the effect will be more pronounced with the increase of the span length and can be weakened in the presence of the axial tension.

Abstract: The nonlinear dunamic behaviors of viscoelastic rectangular plates including the damage effects under the action of a transverse periodic load were studied. Using the von Karman equations, Boltzmann superposition principle and continuum damage mechanics, the nonlinear dynamic equations in terms of the mid-plane displacements for the viscoelastic thin plates with damage effect were derived. By adopting the finite difference method and Newmark method, these equations were solved. The results were compared with the available data. In the numerical calculations, the effects of the external loading parameters and geometric dimensions of the plate on the nonlinear dynamic responses of the plate were discussed. Research results show that the nonlinear dynamic response of the structure will change remarkably when the damage effect is considered.

Abstract: This study investigates the error which occurs when numerically integrating the equation of motion of a single degree of freedom system excited by a harmonic force near resonance. The Constant Average Acceleration method was considered in particular as it features in many finite element software packages. It was found that a considerable error in the calculated responses occurs in systems with low damping due to the well known phenomenon of period elongation. However, the error is reduced for systems with higher damping and/or when smaller time step is used. With regard to this, recommendations are given as to the time steps required to obtain solutions with a pre-defined level of accuracy.

Abstract: At present, most researches on the vortex-induced vibration of submarine free spanning pipelines ignore the effect of internal flowing fluid; furthermore, there are no research reports considering the coupling effect of internal and external fluid with the free span. In this paper, combining Iwan's wake oscillator model with the differential equation derived for the dynamic response of submarine free spanning pipelines with inclusion of internal flow, the pipe-fluid coupling equations are developed to investigate the effect of internal flow on the vortex-induced vibration of the free spans. The finite element approximation is implemented to derive the matrix equations of equilibrium. The Newmark method combined with simple iteration is used to solve the system of equations. The results indicate that the internal fluid flow may cause the shift of resonance band to the lower frequency and a slight decrease in the peak value; the effect will be more pronounced with the increase of the span length and can be weakened in the presence of the axial tension.

Abstract: The behavior of nonlinear vibration for symmetric angle-ply laminated plates including the material viscoelasticity and damage evolution is investigated. By employing the von Karman's nonlinear theory, strain energy equivalence principle and Boltzmann superposition principle, a set of governing equations of nonlinear integro-differential type are derived. By applying the finite difference method, Newmark method and iterative procedure, the governing equations are solved. The effects of loading amplitudes, exciting frequencies and different ply orientations on the critical time to failure initiation and nonlinear vibration amplitudes of the structures are discussed. Numerical results are presented for the different parameters and compared with the available data.

Abstract: The nonlinear transient response of laminated composite shell panels subjected to low velocity impact in hygrothermal environments is investigated using finite element method. The present formulation considers doubly curved thick shells and includes large deformations with Green- Lagrange strains. The analysis is carried out using quadratic eight—noded isoparametric element. A modified Hertzian contact law is incorporated into the finite element program to evaluate the impact force. The nonlinear equation is solved using the Newmark average acceleration method in conjunction with an incremental modified Newton-Raphson scheme. The validity of the present analysis is demonstrated by comparing the present results with the solutions available in the literature. A parametric study is carried out to investigate the effects of the curvature and side to thickness ratios of simply supported composite cylindrical and spherical shell panels.

Abstract: The nonlinear dynamic behaviors of viscoelastic rectangular plates including the damage effects under the action of a transverse periodic load were studied Using the von Karman equations, Boltzmann superposition principle and continuum damage mechanics, the nonlinear dynamic equations in terms of the mid_plane displacements for the viscoelastic thin plates with damage effect were derived By adopting the finite difference method and Newmark method, these equations were solved The results were compared with the available data In the numerical calculations, the effects of the external loading parameters and geometric dimensions of the plate on the nonlinear dynamic responses of the plate were discussed Research results show that the nonlinear dynamic response of the structure will change remarkably when the damage effect is considered

Abstract: The dynamic interaction of vehicle-bridge is studied by using transfer matrix method in this paper. The vehicle model is simplified as a spring-damping-mass system. By adopting the idea of Newmark-b method, the partial differential equation of structure vibration is transformed into a differential equation irrelevant to time. Then, this differential equation is solved by transfer matrix method. The prospective application of this method in real engineering is finally demonstrated by several examples.