Numerical Damping Research Articles

Finite element modeling of erosive wear

Material damage caused by the attack of particles entrained in a fluid system impacting a surface at high speed is called ‘Erosion’. Erosion is a phenomenon that takes place in several engineering applications. It also can be used in several manufacturing process such as abrasive waterjet machining. Erosion is a complex process dependent on particle speed, size, angle of attack as well as the behavior of the eroded material. Extensive experimental results have been reported in the literature on the erosion of different materials. Simulating the erosion process through finite element enables the prediction of erosion behavior of materials under different conditions, which will substitute the need of experimentation, and will enable the identification of constants required for existing analytical models. In this paper, an elasto-plastic finite element (FE) model is presented to simulate the erosion process in 3D configuration. The FE model takes into account numerical and material damping, thermal elastic–plastic material behavior and the effect of multiple particle impacts as well as material removal. The workpiece material modeled was Ti–6Al–4V. The effects of strain hardening, strain rate and temperature were considered in the non-linear material model. Comparison against results reported in literature and erosion models by Finnie, Bitter and Hashish are made. It is shown that the predicted results are in agreement with published results obtained experimentally and from analytical erosion models.

On the Accuracy of Semi-Lagrangian Numerical Simulation of Internal Gravity Wave Motion in the Atmosphere

We have investigated the accuracy ofthe semi-implicit semi-Lagrangian (SISL) method in simulating internal gravity wave (IGW) motion. We have focused on the relative accuracy of the hydrostatic, and nonhydrostatic IGW solutions. The analysis is based on a linearized model and a Global Circulation Model-Dynamic Core (GCM-DC) with a stretched grid.The nonhydrostatic version of the GCM-DC model produces the familiar IGW train disturbance anchored to an isolated hypothetical mountain. The wave has a distinct tilt away from the vertical direction, which is consistent with classical theory. For the hydrostatic version of the model, the axis of the resulting IGW train rests nearly perpendicular to the mountain top, thus again consistent with classical theory. Increasing the time step from 10 s; Courant number (Cn) = 0.5; to 60 s (Cn = 3.0), results in stable solutions for both the hydrostatic and nonhydrostatic versions of the model. The nonhydrostatic solution is in close agreement with the control run however, the hydrostatic solution exhibits large phase truncation errors.The solutions for the one-dimensional linearized SISL model confirm the GCM-DC results that the nonhydrostatic IGW train is less damped and shifted by the SISL scheme than the corresponding hydrostatic IGW motion. The linear solutions indicate very high accuracy of the physical mode of the solution, but it rapidly deteriorates when Cn exceeds unity. As Δt → 0 the amplitude of the computational mode tends to zero and its frequency to infinity. However, as Δt → ∞, the frequency of the computational SISL mode asymptotically approaches the value of the frequency of the corresponding SISL physical mode. Furthermore, the amplitude of the SISL computational mode is directly proportional to the size of the time step. Therefore, at large time steps, the amplification of the computational mode could offset some of the numerical damping of the physical mode by the SISL scheme.

Three-dimensional elastoplastic finite element model for residual stresses in the shot peening process

The fatigue life of mechanical components subjected to variable loading conditions is a crucial parameter in the design of automotive and aerospace mechanical components. The fatigue strength of the component can be improved by inducing compressive residual stress in the surface and subsurface layers by the shot peening process. This increases the failure strain magnitude for crack nucleation and propagation. Extensive experimental results have been published on the effect of shot peening process parameters on residual stress distributions and subsequently on fatigue performance. Numerical simulation of residual stres induced by the shot peening process is important for understanding and optimization of the process in terms of the residual stress profile and maximum shot coverage in the shortest time. In this paper, an elastoplastic finite element (FE) model will be presented to simulate the shot peening process in three-dimensional configuration. The FE model takes into account numerical and material damping, thermal elastic—plastic material behaviour, a rigid or deformable body for the shot and the effects of multiple shot impacts. The workpiece material modelled was AISI 4340 steel at HRC 52 ± 2. The effects of strain hardening, strain rate and temperature were considered in the non-linear material model. Results show that the predicted residual stress profile agreed well with that acquired from published experimental results provided the shot was modelled as a deformable body and material damping was considered.

Modified Newmark family for non‐linear dynamic analysis

AbstractA simple modification of the Newmark scheme is suggested, which keeps the total energy of mechanical systems constant and thereby enhances the unconditional stability in non‐linear dynamic analysis. Numerical damping, which is formulated on an energy basis, is also introduced and the dissipative character of the algorithm is guaranteed in the non‐linear regime. Representative examples in support of such a modified Newmark family are presented. Copyright © 2004 John Wiley & Sons, Ltd.

Alternative time-marching schemes for elastodynamic analysis with the domain boundary element method formulation

This work presents alternative time-marching schemes for performing elastodynamic analysis by the Boundary Element Method. The use of the static fundamental solution and the maintenance of the domain integral associated to the accelerations characterize the formulation employed in this work. It is called D-BEM, D meaning domain. Time response is obtained by employing step-by-step time-marching procedures similar to those adopted in the Finite Element Method. Among all integration procedures, Houbolt scheme became the most popular used to march in time with D-BEM formulation, in spite of the presence of a high numerical damping. In order to improve the integration, this work presents alternative schemes that can be used to perform elastodynamic analysis by the BEM with a better damping control. In order to verify the accuracy of the proposed scheme, three examples are presented and discussed at the end of this work.

The case of an oscillating manometer with variable density and dissipation: experimental and numerical study

With the increasing use of computational fluid dynamics (CFD)-based simulations in the assessment of thermal hydraulic behaviour of nuclear reactors, there is a need to benchmark the predictions of CFD codes. The case of a frictionless U-tube manometer [Ransom, V., 1992. Oscillating manometer. In: Hewitt, G.F., Delhaye J.M., Zuber N. (Eds.), Multiphase Science and Technology, vol. 6. Hemisphere, New York, USA, p. 591] has been previously used as a test case to benchmark the dispersive and dissipative characteristics of a numerical prediction. In the present paper, this case is extended to cover two new aspects that are often found in nuclear reactor systems: variable density and dissipation. Through a theoretical model, it is shown that the dynamic behaviour of the new system is more complicated than that of a constant-density frictionless U-tube manometer. Experiments of the test case have been carried out to substantiate the essential features of the response. CFD simulations of the system show both dissipative and dispersive errors compared to the theoretical result. The results can therefore serve as a useful benchmark for computer codes used to study the thermal hydraulics of nuclear reactor systems, especially in evaluating the role of numerical damping in systems in which physical damping exists inherently.

A mixing-length model for shallow turbulent wakes

A three-dimensional boundary-layer model of shallow-water flows assuming hydrostatic pressure with negligible numerical diffusion and wave damping has been extended to turbulent flow. A standard two-layer mixing-length model determines vertical length scales. The horizontal mixing length is made a multiple and 0.36 were qualitatively reproduced. The flows were subcritical with an onset Froude number of about 0.2, with values approaching 0.6–0.7 in areas where depth-averaged vorticity magnitude was also greatest, at a small distance from the wet/dry intersection. At this intersection, depth-averaged vorticity approached zero while potential vorticity (depth-averaged vorticity/depth) was at a maximum, indicating the importance of the intersection as an origin for vorticity.

Blended Lobatto methods in multibody dynamics

AbstractConvex combination of the L‐stable Lobatto IIIC method and the A‐stable Lobatto IIIA method leads to the new class of blended Lobatto methods. These integration schemes offer the possibility to adapt the numerical damping and asymptotic behaviour to the given tolerance and the stiffness of the multibody system. The paper introduces the method class and discusses advantages and disadvantages of numerical damping. Moreover, a parameter function is given, which optimizes the efficiency by studying the cost of accuracy. As examples, a linear equation and a stiff pendulum are simulated.

A Robust Simulation Algorithm for Conservative Linear Mechanical Systems

Proposed in this article is an inherently conservative simulation scheme, based on the zero-order hold and a closed-form time response of the system at hand. It is shown that the simulation algorithm can be cast in a form in which the state-transition matrix, mapping a state at instant tk into a state at instant tk+1, is proper orthogonal. Hence, for an n-degree-of-freedom (dof) system, this matrix represents a rotation in the 2n-dimensional space of state variables. As a result, the numerical damping required in the simulation of undamped systems is obviated. The performance of the algorithm is illustrated with two examples drawn from engineering systems.

Some Remarks on a Vibro-Impact Scheme

We consider a mechanical system subject to a unilateral constraint for which an impact law is defined. The impact process is considered to be instantaneous and the behavior at the impact is described through a restitution coefficient e. We solve this problem using an algorithm developed and used by Paoli and Schatzman. We show that this algorithm develops numerical instabilities related to the restitution coefficient used in the impact law. Finally, we propose a new scheme which introduces some numerical damping in order to avoid the numerical instabilities.

Application of Runge-Kutta-Rosenbrock Methods to the Analysis of Flexible Multibody Systems

Numerical integration methods are discussed for general equations of motion for multibody systems with flexible parts, which are fairly stiff, time-dependent and non-linear. A family of semi-implicit methods, which belong to the class of Runge–Kutta–Rosenbrock methods, with rather weak non-linear stability properties, are developed. These comprise methods of first, second and third order of accuracy that are A-stable and L-stable and hence introduce numerical damping and the filtering of high frequency components. It is shown, both from theory and examples, that it is generally preferable to use deformation mode coordinates to global nodal coordinates as independent variables in the formulation of the equations of motion. The methods are applied to a series of examples consisting of an elastic pendulum, a beam supported by springs, a four-bar mechanism, and a robotic manipulator with collocated control.

Moment tensors and micromechanical models

A numerical modelling approach that simulates cracking and failure in rock and the associated seismicity is presented and a technique is described for quantifying the seismic source mechanisms of the modelled events. The modelling approach represents rock as an assemblage of circular particles bonded together at points of contact. The connecting bonds can break under applied stress forming cracks and fractures in the modelled rock. If numerical damping is set to reproduce realistic levels of attenuation, then energy is released when the bonds break and seismic source information can be obtained as damage occurs. A technique is described by which moment tensors and moment magnitudes can be calculated for these simulated seismic events. The technique basically involves integrating around the source and summing the components of force change at the surrounding particle contacts to obtain the elements of the moment tensor matrix. The moment magnitude is then calculated from the eigenvalues of the moment tensor. The modelling approach is tested by simulating a well-controlled experiment in which a tunnel is excavated in highly stressed granite while microseismicity is recorded. The seismicity produced by the model is compared to the actual recorded seismicity underground. The model reproduces the spatial and temporal distribution of seismicity observed around the tunnel and also the magnitudes of the events. A direct comparison between the actual and simulated moment tensors is not performed due to the two-dimensional nature of the model, however, qualitative comparisons are presented and it is shown that the model produces intuitively realistic source mechanisms. The ability to obtain seismic source information from the models provides a unique means for model validation through comparison with actual recorded seismicity. Once it is established that the model is performing in a realistic manner, it can then be used to examine the micromechanics of cracking, failure and the associated seismicity and to help resolve the non-uniqueness of the geophysical interpretation. This is demonstrated by examining in detail the mechanics of one of the modelled seismic events by observation of the time dependence of the moment tensor and by direct examination of the particle motions at the simulated source.

TIME INTEGRATION OF NON-LINEAR DYNAMIC EQUATIONS BY MEANS OF A DIRECT VARIATIONAL METHOD

Non-linear dynamic problems governed by ordinary (ODE) or partial differential equations (PDE) are herein approached by means of an alternative methodology. A generalized solution named WEM by the authors and previously developed for boundary value problems, is applied to linear and non-linear equations. A simple transformation after selecting an arbitrary interval of interest T allows using WEM in initial conditions problems and others with both initial and boundary conditions. When dealing with the time variable, the methodology may be seen as a time integration scheme. The application of WEM leads to arbitrary precision results. It is shown that it lacks neither numerical damping nor chaos which were found to be present with the application of some of the time integration schemes most commonly used in standard finite element codes (e.g., methods of central difference, Newmark, Wilson-θ, and so on). Illustrations include the solution of two non-linear ODEs which govern the dynamics of a single-degree-of-freedom (s.d.o.f.) system of a mass and a spring with two different non-linear laws (cubic and hyperbolic tangent respectively). The third example is the application of WEM to the dynamic problem of a beam with non-linear springs at its ends and subjected to a dynamic load varying both in space and time, even with discontinuities, governed by a PDE. To handle systems of non-linear equations iterative algorithms are employed. The convergence of the iteration is achieved by takingn partitions of T. However, the values of T/n are, in general, several times larger than the usual Δt in other time integration techniques. The maximum error (measured as a percentage of the energy) is calculated for the first example and it is shown that WEM yields an acceptable level of errors even when rather large time steps are used.

On the damping analysis of FRP laminated composite plates

This paper presents the damping analysis of fiber reinforced plastics laminated composite plates. For this purpose, the maximum strain and kinetic energies of a cross-ply laminated plate are evaluated analytically based on the three-dimensional theory of elasticity. The displacements of the simply supported rectangular plates are expanded into the polynomial forms with respect to a thickness coordinate, and then governing equations are formulated by using the Ritz's method. In the numerical calculations, natural frequencies and modal damping ratios are calculated for the plates with different stacking sequence and thickness ratios. The validity of the assumption of deformations and the applicability of the other plate theories (e.g. classical lamination theory (CLT), first-order shear deformation theory (FSDT) and higher-order shear deformation theory) to the laminated thick plates are discussed by comparing the numerical results obtained by the present method with the CLT and the FSDT solutions.

A study on time schemes for DRBEM analysis of elastic impact wave

The precise integration and differential quadrature methods are two new unconditionally stable numerical schemes to approximate time derivative with more than the second order accuracy. Recent studies showed that compared with the Houbolt and Newmark methods, they produced more accurate solutions with large time step for the problems where response is primarily dominated by low and intermediate frequency modes. This paper aims to investigate these time schemes in the context of the dual reciprocity BEM (DRBEM) formulation of various shock-excited scalar elastic wave problems, where high modes have important affect on traction response. The Houbolt method was widely recommended in many literatures for such DRBEM dynamic formulations. However, this study found that the damped Newmark algorithm was the most efficient and accurate for impact traction analysis in conjunction with the DRBEM. The precise integration and differential quadrature methods are shown inapplicable for such shock-excited problems due to the absence of numerical damping. On the other hand, we also found that to achieve the same order of accuracy, the differential quadrature method required much less computing effort than the precise integration method due to the use of the Bartels–Stewart algorithm solving the resulting Lyapunov matrix analogization equation.

An energy decaying scheme for nonlinear dynamics of shells

A novel integration scheme for nonlinear dynamics of geometrically exact shells is developed based on the inextensible director assumption. The new algorithm is designed so as to imply the strict decay of the system total mechanical energy at each time step, and consequently unconditional stability is achieved in the nonlinear regime. Furthermore, the scheme features tunable high frequency numerical damping and it is therefore stiffly accurate. The method is tested for a finite element spatial formulation of shells based on mixed interpolations of strain tensorial components and on a two-parameter representation of director rotations. The robustness of the scheme is illustrated with the help of numerical examples.

B-Spline 기저 고차경계요소법에 의한 자유수면하의 2차원 물체주위 유동해석

B-spline을 이용하여 물체의 형상과 포텐셜을 표현함으로써, 저차경계요소법의 단점들을 극복하고 수치계산의 정도를 높이기 위한 고차 패널법을 개발하였다. 물체표면과 자유표면에 법선 다이폴과 쏘스를 분포시켰으며, 자유표면 및 방사조건을 만족시키기 위해 상류차분식을 사용하는 대신 B-spline의 기저함수를 미분하여 선형화된 자유표면 경계조건에 직접 적용하였다. 이 방법을 적용함으로써 Dawson 방법에서 문제가 되었던 수치감쇠 문제를 피할 수 있었다. 수치계산 프로그램을 검증하기 위해 2차원 원주주위의 유동계산과 날개면 주위의 유동해석을 수행하였으며, B-spline 기저 고차패널법에 의한 수치계산 결과가 저차패널법에 비해 빠른 수렴성과 정확성을 보였고 계산에 필요한 패널 수가 현저히 줄어드는 대단히 만족스러운 결과를 얻었다. A two-dimensional higher order panel method using B-splines has been developed to overcome the disadvantages of the low order panel method and to obtain more accurate solution. The sources and the normal dipoles are distributed on both the body and the free surface. Instead of applying the upwind finite difference schemes to satisfy the linearized free surface and the radiation condition, the derivatives of the basis functions of the B-splines are directly applied to the linearized free surface condition. Numerical damping in the Dawson's method are avoided in the Present computations. In order to validate the present method, numerical computations are carried out for a submerged cylinder and a two-dimensional hydrofoil steadily moving beneath a free surface. The numerical results show that fast convergence and better accuracies have been achieved by the present method.

The Study of Analytical Models for Vibration of Cross-ply Laminated Thick Plates.

This paper studies the assumption of displacements employed in the vibration analysis of laminated composite plates. For this purpose, the maximum strain and kinetic energies of a cross-ply laminated plate are evaluated analytically based on the three-dimensional theory of elasticity. The displacements of the simply-supported rectangular plates are expanded into the polynomial forms with respect to a thickness coordinate, and then governing equations are formulated by using the Ritz's method. In the numerical calculations, natural frequencies and modal damping ratios are calculated for the plates with different stacking sequence and the thickness ratios. The validity of the assumption of displacements and the applicability of the other plate theories (e. g. Classical Lamination Theory, First-order Shear Deformation Theory and Higher-order Shear Deformation Theory) to the laminated thick plates are discussed by comparing the numerical results obtained by the present method with the CLT and the FSDT solutions.

Hybrid Finite-Difference Scheme for Solving the Dispersion Equation

An efficient hybrid finite-difference scheme capable of solving the dispersion equation with general Peclet conditions is proposed. In other words, the scheme can simultaneously deal with pure advection, pure diffusion, and/or dispersion. The proposed scheme linearly combines the Crank-Nicholson second-order central difference scheme and the Crank-Nicholson Galerkin finite-element method with linear basis functions. Using the method of fractional steps, the proposed scheme can be extended straightforwardly from one-dimensional to multidimensional problems without much difficulty. It is found that the proposed scheme produces the best results, in terms of numerical damping and oscillation, among several non-split-operator schemes. In addition, the accuracy of the proposed scheme is comparable with a well-known and accurate split-operator approach in which the Holly-Preissmann scheme is used to solve the pure advection process while the Crank-Nicholson second-order central difference scheme is applied to the pure diffusion process. Since the proposed scheme is a non-split-operator approach, it does not compute the two processes separately. Therefore, it is simpler and more efficient than the split-operator approach.

Stability improvement to BEM/FEM coupling scheme for 2D scalar wave problems

The stability problem appeared in boundary element method/finite element method (BEM/FEM) coupling is discussed in this paper. As the response at time t n+1 relates to the excitations and responses at all previous times, i.e. response history, BEM is easier to be unstable compared with FEM. The Newmark method for FEM is unconditionally stable, oscillations appeared at any time would decrease step by step as time goes by. But the oscillation history caused by FEM may be big enough to cause stability problems to the BEM scheme which although may be stable when used independently. A new procedure is used in this paper to reduce the oscillation history caused by FEM so that it will not cause stability problem to BEM scheme and further to the coupling BEM/FEM scheme. Numerical examples show that the proposed procedure can improve significantly to the stability of the coupling BEM/FEM scheme and cause little numerical damping.