Time Step Increment Research Articles

Successive linear approximation for boundary value problems of nonlinear elasticity in relative-descriptional formulation

A method of successive linear approximation in current configuration for solving boundary value problems of large deformation in finite elasticity is proposed. Instead of using Lagrangian or Eulerian formulation, we can also formulated the problem relative to the current configuration, and linearize the constitutive function at the present state so that it leads to a linear boundary value problem for an incremental time step. Therefore, as linearization at present state proceed in time, problem for large deformation can be formulated. The idea is similar to the Euler’s method for differential equations. As an example for the proposed method, numerical simulation of bending a rectangular block into a circular section for Mooney–Rivlin material is given for comparison with the exact solution, which is one of the well-known universal solutions in finite elasticity.

Dynamic analysis methodology for progressive failure of truss structures considering inelastic postbuckling cyclic member behavior

Over the years, several catastrophic collapses of truss structures have been reported. Sudden failure or reduction in member capacity of a single member in a truss structure gives rise to dynamic force redistribution in the remaining members and may lead to progressive collapse of the entire structure. During failure, truss members can undergo inelastic cyclic behavior (including postbuckling in compression and yielding in tension) that may not have existed in the intact structure. This paper presents a methodology to incorporate the inelastic cyclic member force–deformation behavior in the dynamic analysis of truss structures and at the same time incorporates the possible dynamic effects arising from the sudden change in load carrying capacity of a member due to failure or buckling/postbuckling. The method tracks and generates the force–deformation characteristics of every member of the truss at each incremental time step. The continuous change in the load-carrying capacity and the stiffness of members during the nonlinear force–deformation history has been incorporated in the analysis scheme using the Pseudo-force approach. The solution methodology for obtaining the dynamic response of the structure is based on the finite element technique and considers elasto-plastic material and large deformation geometric nonlinearities. The methodology is applied to a two-dimensional three-member toggle redundant truss subjected to external static, quasi-static, and dynamic (sinusoidal and ramp) loads. Results delineating the effects of the inelastic cyclic axial force–deformation relation of each member and the time variation of joint displacements and member forces are presented for each loading condition. The results show that there exist cases where modeling a compression member with its actual postbuckling behavior, which although has some reserve load carrying capacity, are more critical than the case where the same member is considered to suddenly lose its full load carrying capacity at its buckling load.

Transient Response Analysis of a High Speed Rotor Using Constraint Real Mode Synthesis

In order to accurately calculate high-speed nonlinear transient phenomena of a large order rotor system such as rotor collision and rubbing with stator repeatedly due to blade loss, a very small incremental time step is required in numerical integration. But a small time step is known to become a cause of divergence or flutter type numerical instability when a direct integration such as Newmark-β method is applied in terms of physical coordinates. Normal mode method, which uses eigen modes instead of physical freedoms, is effective on this numerical stability, but the modes in the calculation are usually complex-modes and also need recalculation in accordance with varying rotational speed. Therefore, this method is inconvenient for an analysis of a rotor, especially speed varying system. This paper presents a new method using constraint real mode synthesis in which rotor modes are derived by neglecting gyroscopic moment as at 0 rpm and by fixing bearing connection interface freedoms to zero. This method is proved to be highly stable and also to have good accuracy and efficiency in the calculation.

Long-term effects due to creep and shrinkage on prestressed concrete bridge girders using SCC

Prestressed concrete structures are sensitive to long-term deformations due to creep and shrinkage. Over estimation of the camber due to these effects lead to inefficient designs and improper utilisation of the material strength. In this contrast this study aims at analysing the long-term behaviour of self-compacting concrete (SCC) in prestressed concrete bridge girders. The SCC mix is designed and the hardened properties are tested. Design strength of 71 MPa is obtained at 28 days of curing. A suitable model is selected from various available models presented in the literature to estimate the creep and shrinkage characteristics of the design mix of SCC. The study is furthered to predict the camber deformations in prestressed concrete Bridge girders. A case study of the Alabama Bridge is selected, for which the measured results of camber deformations are available. An incremental time step procedure is followed, incorporating the creep and shrinkage characteristics of the design mix. The outcome of the study predicts lower camber deformations for the design mix of concrete in the bridge girders.

Perfect plane wave injection for Crank–Nicholson time-domain method

The authors derive a 1D multipoint auxiliary source propagator for perfect plane wave injection (PWI) with the Crank–Nicholson time-domain scheme for the first time. By projecting the 2D dispersion equation onto the 1D case, the identical dispersion relation can be realised between the 1D case and the 2D case, which leads to a perfect PWI at any angle forming an integer grid cell ratio. Several numerical experiments are conducted, which show leakage errors on the order of finite precision (−300 dB for double precision) when Δt is around the Courant–Friedrichs–Levy (CFL) limit. Beyond the CFL limit, the leakage error is still kept low even though it deteriorates with increasing time-step increments. Good agreement is observed between the proposed method and analytic solution for scattering of a 2D perfect electric conductor circular scatterer.

Numerical simulation of bubble growth in a limited amount of liquid

AbstractIn this study, we examined the growth of a spherical bubble in a limited amount of liquid by using a finite‐element‐based numerical simulation method. The bubble growth was assumed to be controlled by both momentum and mass transfer. A truncated power‐law constitutive equation was used to describe the rheology of the melt. The gas inside the bubble followed the ideal gas law. The gas concentration at the bubble surface obeyed Henry's law. A computer code was programmed to solve the equations with the Galerkin method. A backward Euler scheme was used to discretize time. Grids were remeshed after each incremental time step to ensure the accuracy of the numerical results. The bubble growth process was simulated with the code. The numerical results, such as the instantaneous bubble size, gas pressure inside the bubble, and gas concentration profile in the liquid, were predicted. The influences of the liquid volume, initial gas pressure, temperature, and rheology of the melt on bubble growth were also studied. The results of the bubble growth simulation in this study were in satisfactory agreement with others' work. © 2009 Wiley Periodicals, Inc. J Appl Polym Sci, 2010

A FUNDAMENTAL STUDY ON APPLICABILITY OF DISCONTINUOUS DEFORMATION ANALYSIS TO MASONRY ARCHITECTURE

In this paper, we study on the applicability of Discontinuous Deformation Analysis (DDA) to dry-masonry structures such as Angkor. Many parameters are needed for DDA and we used contact force as a precision index to set guidelines for some parameters, i.e. close-line setting inside of a block, penalty spring constant, and time step increment. We also measured predominant frequency of block systems composed of different number of blocks and simulated them by DDA. We found that a system of different number of blocks requires different penalty spring constant. With these fundamental studies, we are elucidating collapse mechanism of dry-masonry structures.

実数拘束モード合成法による高速回転軸系の過渡応答解析(機械力学,計測,自動制御)

In order to calculate correctly high-speed nonlinear transient phenomena of a large order rotor system such as rotor collision and rubbing with stator repeatedly due to blade loss, a very small incremental time step is required in numerical integration. But a small time step is known to become a cause of divergence or flutter type numerical instability when a direct integration such as Newmark-β method is applied in terms of physical coordinate. Normal mode method, which uses eigen modes instead of physical freedoms, is effective on this numerical stability, but the modes in the calculation are usually complex modes and also need recalculation in accordance with varying rotational speed. Therefore, this method is inconvenient for an analysis of a rotor especially speed varying system. This paper presents a new method using constraint real mode synthesis in which rotor modes are derived by neglecting gyroscopic moment as at 0rpm and by fixing bearing connection interface freedoms to zero. This method is proved to be highly stable and also to have good accuracy and efficiency in the calculation.

Comparative modeling of stand development in Scots pine dominated forests in Estonia

In general, forests in Estonia are characterized by great variability, not only in protected areas but in commercial forests as well. The data needed for the derivation and calibration of growth models can be obtained by continuous observation of permanent growth plots (also known as longitudinal studies) or by establishing chronosequences with temporary plots distributed over a wide range of growing sites, densities and ages (also known as cross-sectional studies). A compromise may be achieved by a system of “interval plots” (also known as a short-time series: series which covers a short time). Since the measurement interval is a period of undisturbed growth, it is possible to measure change rates as in a longitudinal study and at the same time cover a wide range of initial conditions as in a cross-sectional study. Numerous models of stand growth have been derived from re-measured sample plots. This study, which uses the data of 142 five-year intervals from 134 unmanaged Scots pine stands, compares six different model combinations involving algebraic difference equations and fixed time-step increment equations. New stand-level diameter and basal area increment equations and a tree survival model which showed close correspondence with the existing stand-level model for Estonia were developed. The main advantage of the use of algebraic difference equations over the fixed-step increment equations is the ability to use flexible time steps. However, the projection intervals should not deviate too much from the time steps of the measurement data. An important constraint when using the algebraic difference equations is to avoid long-term predictions in one projection sequence.

A NEW TECHNIQUE ESTIMATING LOCATION OF MEAN JOINT CENTERS OF ROTATION WITH ASSOCIATED DISPERSIONS AND ASSESSING TO FUNCTIONAL COORDINATE SYSTEM OF MOVING LIMB SEGMENTS

Joint centers are obtained from data treatment of a set of markers placed on the skin of moving limb segments. Finite helical axis (FHA) parameters are calculated between time step increments. Artifacts associated with nonrigid body movements of markers entail ill-determination of FHA parameters. Mean centers of rotation may be calculated over the whole movement, when human articulations are likened to spherical joints. They are obtained using numerical technique, defining point with minimal amplitude, during joint movement. A new technique is presented. Hip, knee, and ankle mean centers of rotation are calculated. Their locations depend on the application of two constraints. The joint center must be located next to the estimated geometric joint center. The geometric joint center may migrate inside a cube of possible location. This cube of error is located with respect to the marker coordinate systems of the two limb segments adjacent to the joint. Its position depends on the joint and the patient height, and is obtained from a stereoradiographic study with specimen. The mean position of joint center and corresponding dispersion are obtained through a minimization procedure. The location of mean joint center is compared with the position of FHA calculated between different sequential steps: time sequential step, and rotation sequential step where a minimal rotation amplitude is imposed between two joint positions. Sticks are drawn connecting adjacent mean centers. The animation of stick diagrams allows clinical users to estimate the displacements of long bones (femur and tibia) from the whole data set.

Modeling of turbulent heat transfer during the solidification process of continuous castings

Prandtl mixing length theory with average heat capacity was successfully implemented to model turbulent continuous casting (CC) process. This method features the simplicity of zero equation model along with ease of maintaining higher time step increment using average heat capacity method. A series of simulations were investigated to study the effect of inlet speed ( Pe), superheat ( Θ o), mold cooling rate ( Bi m) and post mold cooling rate ( Bi p) on the process characteristics and were presented in terms of viscosity contours, location and slope of the solidification front, axial velocity profiles, velocity vectors, local heat flux and shell thickness. The ranges of the parameters studied in the current work were: Pe = 3.5–6.1, Θ o = 1.2–2.0, Bi m = 0.2–0.4 and Bi p = 0.1–0.4. The effect of mold length ( L m) was also investigated to establish remedies of breakout due to higher Pe and Θ o. The current studies reveal that Pe, Θ o and L m have significant effect on the performance and productivity of the cast material.

Numerical performance and applications of the envelope ADI-FDTD method

The numerical performance of the envelope alternating-direction-implicit-finite-difference time-domain (ADI-FDTD) method and its applications are studied in this paper. The ADI-FDTD method is independent of the Courant-Friedrich-Levy stability condition, but its numerical dispersion grows with the increase of the time-step size. By introducing the envelope technique in the ADI-FDTD method, the numerical accuracy can be improved efficiently. In this paper, the phase velocity error of a propagating Gaussian pulse was studied for the envelope ADI-FDTD and ADI-FDTD and conventional FDTD methods with different cell size and time-step increment, then two waveguide problems and a scattering problem were simulated with the envelope ADI-FDTD and ADI-FDTD methods in graded meshes and the conventional FDTD method in a uniform mesh. The simulation results show the superior performance of the envelope ADI-FDTD over the ADI-FDTD in numerical accuracy.

331 実数拘束モード合成法による高速回転軸系の過渡応答解析

In order to calculate high-speed transient phenomena of a rotor-system such as rotor collision with stator due to blade loss, a very small incremental time step is required in the integration. But a small time step is known to cause a numerical instability when a direct integration such as Newmark-β method is applied in terms of physical coordinate. This paper presents a method using constraint real mode synthesis in which modes are derived from neglecting gyro-moment and fixing bearing connection freedoms to be zero. This method is proved to be highly stable and also to have good accuracy and efficiency.

Enlarged cells for the conformal FDTD method to avoid the time step reduction

Recently, the conformal finite-difference time-domain (CFDTD) method has emerged as an efficient FDTD method with a higher order accuracy than the conventional FDTD methods that are degraded by staircasing errors. The only obvious point to further improve on the CFDTD method is its requirement for a smaller time step increment due to the existence of small irregular cells near the boundary. In this letter, an enlarged cell technique is introduced to ensure the stability of the CFDTD method without the time step reduction. The introduction of the enlarged cells therefore makes the CFDTD method much more efficient and suffers from a smaller dispersion error, as shown in several two-dimensional examples.

NUMERICAL INVESTIGATION OF THE EFFECTS OF TURBULENCE FROM SUBMERGED ENTRY NOZZLE DURING CONTINUOUS CASTING PROCESS

A numerical investigation was performed using Prandtl mixing-length theory along with the average heat capacity method to model a turbulent plane submerged entry nozzle continuous casting process. This method features the simplicity of a zero-equation model along with ease of maintaining the time-step increment. Results from the study match well with other published data. The current study indicates that submergence depth of the entry nozzle affects the turbulence level and hence controls the quality of the finished product. The inlet speed of the molten metal also controls the solidification front slope and the surface heat flux of the caster.

Algorithm and implementation of soil–tire contact analysis code based on dynamic FE–DE method

One of the fundamental problems in terramechanics is soil–tire system. Past achievements on this topic can be observed in various literatures. Fast development on CPU power of PC system enables us to apply numerical methods to this basic subject. Among others, finite element method (FEM) has been applied to simple problems of soil–tire system not only in 2D but also in 3D approach. However, it is noted that the current FEM technology cannot handle “singular” boundary conditions with sufficient accuracy of analysis. Typical example of this limitation can be seen in an application to traction tire–soil contact problems, where the contact point of tire lug tip behaves as the singular point of stress field. On the other hand, distinct or discrete element method (DEM) has in essence the capability of analyze microscopic deformation (or flow) of soil as many researchers have already been demonstrated. It is noted that DEM suffers large calculation time that is consumed not only at contact check between particulate elements but also at incremental time step. In our present study, we try to combine both merit of FEM and DEM together in order to analyze the soil–tire system interaction, where, for example, a tire and deep soil layer are modeled as FEM and soil surface layer as DEM. We propose simple algorithm of this FE–DE coupled method and sample program is developed that can solve some basic terramechanics problems in order to verify our idea. The obtained result shows qualitatively sufficient accuracy.

Aeroelastic Wind Forces on Flexible Bridge Girders

Non-stationary aeroelastic wind force processes on a streamlined bridge deck are modelled for use within computations in the range of statistically stationary girder oscillation and for the mathematical investigation of the limit of aeroelastic stability. Time Domain (TD) methods, as the direct integration on a finite-element (FE) model, are applied in structural analyses for taking into consideration both geometric and physical nonlinearities. The FE method requires a continuous form of the force model, which permits an arbitrary discretization in time. The present model is based on continuous step response functions of Wagner-type, taking advantage of modern advances of that theory. Special emphasis is put on the continuity and the existence of equivalent representations in time and frequency domains. A realistic wind load model must include measured force functions and effects from the wind turbulence. For that reason, the force functions are calibrated in wind tunnel tests on section models in grid turbulence and boundary layer flow. The aeroelastic in-phase and out-of-phase forces are identified from highly accurate calibrations experiments on section models up to high frequencies which permits to decrease the time step increment. Turbulence effects on the aeroelastic coefficients are included in mean terms.

Evaluation of Stored Energy in Ultrafine Aluminum Powder Produced by Plasma Explosion

Ultrae ne aluminum powder produced by plasma explosion (ALEX) exhibits burn behavior unlike that of ordinary aluminum powders. Others have previously suggested that the source of this unique behavior might be stored internal energy. The objective of this study is to evaluate theoretically the feasibility of energy being stored in ALEX as a result of work performed by means of the compression of the liquid portion of an ALEX particle by its shrinking solid shell during rapid solidie cation of the particle. A theoretical model of this process of energy storage was developed. This model was then formulated and numerically solved on a computer for a variety of conditions. Results show that the postulated mechanism results in measurable stored energy for only unrealistically high cooling rates. It is concluded that, although the postulated mechanism could store energy, the amount of energy stored is realistically negligible. This theoretical e nding is in agreement with recent experiments that show no observable stored energy in ALEX particles. Nomenclature A = surface area, m 2 B = bulk modulus, N/m 2 Cp = specie c heat, J/kg ¢K D = diameter, m Est = thermal energy stored during solidie cation, J H f = latent heat of fusion, J/kg h = heat transfer coefe cient, W/m 2 ¢K k = thermal conductivity, W/m ¢K m = mass, kg N = surface node P = pressure, Pa r = radius,m T = temperature, K Ti = initial temperature, K V = volume, m 3 W = work, J ¯ = thermal coefe cient of expansion, 1/K 1t = time-step increment, s Ω = density, kg/m 3

The Lattice Solid Model to Simulate the Physics of Rocks and Earthquakes: Incorporation of Friction

The particle-based lattice solid model developed to study the physics of rocks and the nonlinear dynamics of earthquakes is refined by incorporating intrinsic friction between particles. The model provides a means for studying the causes of seismic wave attenuation, as well as frictional heat generation, fault zone evolution, and localisation phenomena. A modified velocity–Verlat scheme that allows friction to be precisely modelled is developed. This is a difficult computational problem given that a discontinuity must be accurately simulated by the numerical approach (i.e., the transition from static to dynamical frictional behaviour). This is achieved using a half time step integration scheme. At each half time step, a nonlinear system is solved to compute the static frictional forces and states of touching particle-pairs. Improved efficiency is achieved by adaptively adjusting the time step increment, depending on the particle velocities in the system. The total energy is calculated and verified to remain constant to a high precision during simulations. Numerical experiments show that the model can be applied to the study of earthquake dynamics, the stick–slip instability, heat generation, and fault zone evolution. Such experiments may lead to a conclusive resolution of the heat flow paradox and improved understanding of earthquake precursory phenomena and dynamics.

Simulated wave propagation in elastic environmental flat and sloping waveguides by a finite length ping time-dependent ray trace model

A finite length ping ray trace program (FLPRT) has been extended that is now capable of simulating time-dependent wave motion in a complicated elastic bathymetric waveguide. From the original concept of sound waves being generated from an infinite set of poiny sources, FLPRT expands a finite set into following ray theory including shear and compression logic in a half space. Wave propagation is demonstrated in a constant velocity shallow ocean environment for both flat and sloping bathymetries. Not only is the objective to compare the model-generated transmission loss calculations against those computed by EFEPE, an elastic parabolic equation model by M. Collins of ARL, but also to show the time-generated path of the acoustic waves in both cases. Narrow angle generated pings show different nonmodal characteristics than those generated by wide angle pulses which exhibit fluctuating waves predicted by normal-mode theory. As in all cases, adiabaticity is assumed within each very small incremental time step. A 3–5 min VCR(TV) presentation will be shown demonstrating wave motion in the presence of a flat and wedge half-space exhibiting both shear and compressional waves on the half-space.