Overcoming Load Discontinuity in Step-by-Step Solution of Shock Response

A scheme is proposed to overcome the discontinuity at the end of an impulse. This scheme is very simple in the step-by-step solution of shock response. The only change is the loading input at the time instant of load discontinuity in performing the step-by-step integration. The average value of the two discontinuity values at the integration point of load discontinuity is used to replace the use of one of them for loading input. The motivation of this change originates from the concept of no loading input error associated with the integration point of load discontinuity. The feasibility of this scheme is analytically explored. Analytical results reveal that this change in loading input will lead to no extra impulse and thus no extra displacement. Consequently, an accurate shock response can be computationally efficiently obtained. Two numerical examples are also used to confirm the analytical results. The scheme proposed will not increase any computational effort or complicate the dynamic analysis codes.     

A problem-based education program for patients with rheumatoid arthritis: evaluation after three and twelve months

OBJECTIVES: The aim of this work was to present a preliminary numerical analysis of the integration process of dental implants using a finite element simulation of the dynamic response following impulse excitation. Assessment of the osseointegration process has been previously examined using a numerical approach by calculating the natural frequency of a cantilever attached to the implant. The methodology adopted in this work allows a direct measurement of the implant response following impulse loading and avoids the addition of a bulky cantilever set-up. METHODS: The geometric configuration was obtained by averaging the coordinate data from tomographic scans of 14 mandibles. The materials properties were approximated from experimental analysis performed on trabecular and cortical bone tissue. A load was applied to the top of the implant in one direction resulting in an initial displacement. The implant was then freed and allowed to vibrate over approximately 10 cycles. Three fixity conditions were assumed by changing the properties of the surrounding bone ranging from full integration to a poorly integrated implant typical of the situation during bone healing following surgery. The results of the three fixity conditions were compared by calculating the fundamental displacement amplitudes and frequencies of the vibrating impact. RESULTS: The calculated results indicated that the implant vibrated at a predominant frequency when partially integrated with a displacement principally in the direction of the applied impulse. However, when the implant was fully integrated a more complex vibration pattern ensued, suggesting the superposition of two or more fundamentals. SIGNIFICANCE: Attention has been paid to the formulation of the numerical model for validation purposes as well as a reliable reference for the optimum interpretation of the experimental data. In this way it was possible to establish a simulation procedure to investigate the response of the tissues surrounding the implant and their properties at different stages of healing. It should be pointed out that the numerical procedures represented a valid preliminary approach to the problem and were capable of indicating a guide to the optimum design of the experimental apparatus for measurement of displacement and frequency in vivo.     

Dynamic Behavior of Reinforced Concrete Beams Strengthened with Composite Materials

The dynamic behavior of reinforced concrete (RC) beams strengthened with externally bonded composite materials is analytically investigated. The analytical model is based on dynamic equilibrium, compatibility of deformations between the structural components (RC beam, adhesive, composite material) and the concept of the high order approach. The equations of motion along with the boundary and continuity conditions are derived using Hamilton’s variational principle and the kinematic relations of small deformations. The mathematical formulation also includes the constitutive laws that are based on beam and lamination theories, and the two-dimensional elasticity representation of the adhesive layer including the closed form solution of its stress and displacement fields. The Newmark time integration method, which is directly applied to the resulting set of coupled partial differential equations, is adopted. This procedure yields a set of ordinary differential equations, which are analytically or numerically solved in every time step. The response of a strengthened beam to different dynamic loads that include impulse load, harmonic load, and seismic base excitation is numerically investigated. The numerical study highlights some of the phenomena associated with the dynamic response and explores the capabilities of the proposed model. The paper closes with a summary and conclusions.     

Enhanced, Unconditionally Stable, Explicit Pseudodynamic Algorithm

In performing a pseudodynamic test, an explicit method is generally preferred over an implicit method since it involves no iteration procedure or extra hardware that is often needed for an implicit method. However, its integration time step is usually limited by stability. Hence, it is very promising for the pseudodynamic testing if an explicit method can have unconditional stability, which might eliminate the limitation on time step for the testing of a multiple degree of freedom system or a substructure system. Although an explicit pseudodynamic algorithm with unconditional stability has been successfully implemented and its superior characteristics have been identified, an enhanced unconditionally stable explicit pseudodynamic algorithm is further proposed. In this study, it is verified that both explicit pseudodynamic algorithms possess the same numerical characteristics in the step-by-step integration. However, the newly developed explicit pseudodynamic algorithm shows better error propagation properties when compared to that developed previously.     

Numerical Study of Scale Effects on the Stiffness Modulus of Rock Masses

The purpose of this paper is to contribute to the study of scale effects on the stiffness modulus of discontinuous rock masses from a mechanical point of view. The study has been conducted on the basis of the observation that the length and density of the natural discontinuities (cracks) in rock masses increase with the volume of the mass. Compressive tests on specimens containing open and closed flat cracks have been simulated through a numerical procedure that is based on the displacement discontinuity method and the fictitious stress method. Cracks have been generated, through a random process, with dips in the 0°–180° range. Both open and closed cracks have been considered. The numerical results obtained for specimens containing open cracks are found to approach the values obtained through an analytical solution. Then, the numerical method has been applied to the study of scale effects on the stiffness modulus of specimens with closed cracks. The numerical results show a decrease in the stiffness modulus with an increase in the size of the rock volume.     

Accurate Representation of External Force in Time History Analysis

It is usually thought that the integration time step should be small enough to represent properly the variation of the dynamic loading with respect to time. However, there is no evaluation criterion that can be used to determine whether the external force is accurately represented. In this paper, criteria for accurate representation of external force are proposed based on analytical results. It is found that amplitude distortion both in the transient response and the steady-state response for each time step is closely related to the step discretization error of external force. In fact, for a negligible period distortion, an amplitude distortion will be less than 5% if the relative step discretization error is constrained to be less than 5% at each time step for the Newmark explicit method, Fox–Goodwin method, and linear acceleration method while for the constant average acceleration method it must be less than 2.5%. This criterion leads to the need of using of eight or more integration time steps to accurately represent a complete cycle of a harmonic loading for the Newmark explicit method, Fox–Goodwin method, and linear acceleration method while for the constant average acceleration method 12 or more integration time steps are required.     

Lagrangian Approach to Structural Collapse Simulation

Computer analysis of structures has traditionally been carried out using the displacement method combined with an incremental iterative scheme for nonlinear problems. In this paper, a Lagrangian approach is developed, which is a mixed method, where besides displacements, the stress resultants and other variables of state are primary unknowns. The method can potentially be used for the analysis of collapse of structures subjected to severe vibrations resulting from shocks or dynamic loads. The evolution of the structural state in time is provided a weak formulation using Hamilton’s principle. It is shown that a certain class of structures, known as reciprocal structures, has a mixed Lagrangian formulation in terms of displacements and internal forces. The form of the Lagrangian is invariant under finite displacements and can be used in geometric nonlinear analysis. For numerical solution, a discrete variational integrator is derived starting from the weak formulation. This integrator inherits the energy and momentum conservation characteristics for conservative systems and the contractivity of dissipative systems. The integration of each step is a constrained minimization problem and it is solved using an augmented Lagrangian algorithm. In contrast to the traditional displacement-based method, the Lagrangian method provides a generalized formulation which clearly separates the modeling of components from the numerical solution. Phenomenological models of components, essential to simulate collapse, can be incorporated without having to implement model-specific incremental state determination algorithms. The state variables are determined at the global level by the optimization method.     

Address-Oriented Impedance Matrix Method for Generic Calibration of Heterogeneous Pipe Network Systems

The generic evaluation of pipeline parameters is one of the most demanding technological tasks in the efficient management of a water distribution system. Information about current pipeline status is feasible by monitoring the pressure variation online. Conventional methods of transient computation and parameter calibration for a heterogeneous pipeline network suffer from cost issues both in time and storage as well as several other constraints associated with the numerical representation of a real-life system. As an alternative approach, an extension of the impulse response method, namely the address-oriented impedance matrix method (AOIMM), has been developed for a more robust calibration of a heterogeneous and multilooped pipe network system. The genetic algorithm was incorporated into the AOIMM for generic calibration of several parameters, such as the location and quantity of leakage, friction factor, and wave propagation speed. The potential of the proposed calibration algorithm over other conventional approaches was demonstrated when it was applied to a hypothetical heterogeneous pipe network system.     

Temperature Prediction of Concrete-Filled Rectangular Hollow Sections in Fire Using Green’s Function Method

An efficient numerical approach using the Green’s function solutions of transient heat conduction for predictions of thermal response inside a concrete-filled rectangular hollow section subjected to fire is proposed in this paper. Thermal properties of construction materials are assumed to be isotropic and homogeneous. The Green’s function approach adopts different series expansions for small and large time solutions, therefore the desirable convergence properties can be achieved at any range of time by using the time partitioning strategy. A useful analytical relation in terms of step Green’s functions is derived in this paper to incorporate the multidimensional effect, in particular, for Neumann (prescribed flux) boundary conditions. A modified lumped capacitance method, together with an “orthogonal flux” concept, are employed to deal with spatially varying heat flux at the steel–concrete interface, where Duhamel’s theorem is applied in piecewise manner along the interface to incorporate the fire boundary conditions. No spatial discretization is required in the numerical algorithms based on the Green’s function approach.     

Efficient integration of a realistic two-dimensional cardiac tissue model by domain decomposition

The size of realistic cardiac tissue models has been limited by their high computational demands. In particular, the Luo-Rudy phase II membrane model, used to simulate a thin sheet of ventricular tissue with arrays of coupled ventricular myocytes, is usually limited to 100 x 100 arrays. We introduce a new numerical method based on domain decomposition and a priority queue integration scheme which reduces the computational cost by a factor of 3-17. In the standard algorithm all the nodes advance with the same time step delta t, whose size is limited by the time scale of activation. However, at any given time, many regions may be inactive and do not require the same small delta t and consequent extensive computations. Hence, adjusting delta t locally is a key factor in improving computational efficiency, since most of the computing time is spent calculating ionic currents. This paper proposes an efficient adaptive numerical scheme for integrating a two-dimensional (2-D) propagation model, by incorporating local adjustments of delta t. In this method, alternating direction Cooley-Dodge and Rush-Larsen methods were used for numerical integration. Between consecutive integrations over the whole domain using an implicit method, the model was spatially decomposed into many subdomains, and delta t adjusted locally. The Euler method was used for numerical integration in the subdomains. Local boundary values were determined from the boundary mesh elements of the neighboring subdomains using linear interpolation. Because delta t was defined locally, a priority queue was used to store and order next update times for each subdomain. The subdomain with the earliest update time was given the highest priority and advanced first. This new method yielded stable solutions with relative errors less than 1% and reduced computation time by a factor of 3-17 and will allow much larger (e.g., 500 x 500) models based on realistic membrane kinetics and realistic dimensions to simulate reentry, triggered activity, and their interactions.     

Improved Explicit Method for Structural Dynamics

An explicit method, which simultaneously has the most promising advantages of the explicit and implicit methods, is presented. It is shown that numerical properties of the proposed explicit method are exactly the same as those of the constant average acceleration method for linear elastic systems. However, for nonlinear systems, it has unconditional stability for an instantaneous stiffness softening system and conditional stability for an instantaneous stiffness hardening system. This conditional stability property is much better than that of the Newmark explicit method. Hence, the proposed explicit method is possible to have the most important property of unconditional stability for an implicit method. On the other hand, this method can be implemented as simply as an explicit method, and hence, possesses the most important property of explicit implementation for an explicit method. Apparently, the integration of these two most important properties of explicit and implicit methods will allow the proposed explicit method to be competitive with other integration methods for structural dynamics.     

Comparison between Models of Rock Discontinuity Strength and Deformation

One important component in the design of tunnels in urban areas is a correct assessment of the interaction between the underground excavation with other structures in the vicinity. In this sense a correct stress-strain response by the model representing the rock mass behavior is essential. The shear and normal displacement of rock discontinuities and their shear and normal stiffness control the distribution of stress and displacement within a discontinuous rock mass. In conditions where an equivalent continuum based approach is not applicable, the joint material model should be able to describe important mechanisms such as asperity sliding and shearing, post-peak behavior, asperity deformation, and the effect of soft infilling. The distinct element code UDEC was used to simulate the direct shear tests on a natural joint profile, and the prediction of two existing models of discontinuity strength and deformation were then compared with a new soil-infilled joint model and with experimental data for clean and soil-infilled rock joints. A numerical modeling of a cavern excavated in a jointed medium is also presented to illustrate the response of different models. The proposed soil-infilled joint model described more comprehensively the occurrence of dilation and compression with lateral displacements and also better represented the double peak shearing in relation to the adopted squeezing mechanism that could not be captured by the two existing models.     

Optimizing Aquifer Parameters from Drawdowns in Large Diameter Wells

An optimization method is proposed for estimating the storage coefficient and transmissivity of an aquifer from drawdowns in large diameter wells consequent to an unsteady pumping. The concept of an optimal time step size is propagated in the proposed method. The estimate of the aquifer parameters corresponding to the optimal time step size is termed final estimate. The estimates for any other time step size are not reliable. The proposed method can also take into account the residual drawdowns. The application of the method is illustrated using an example.     

Bridge Pier Scour under Flood Waves

The effect of a single-peaked flood wave on pier scour is investigated both theoretically and experimentally. The conditions considered involve clear-water scour of a cohesionless material of given median sediment size and sediment nonuniformity, an approach flow characterized by a flow depth and velocity, a circular-shaped cylindrical bridge pier, and a flood hydrograph defined by its time to peak and peak discharge. A previously proposed formula for scour advance under a constant discharge was applied to the unsteady approach flow. The generalized temporal scour development along with the end scour depth are presented in terms of mainly the densimetric particle Froude number based on the maximum approach flow velocity and the median sediment size. The effect of the remaining parameters on the end scour depth is discussed and predictions are demonstrated to be essentially in agreement with model observations.     

Porothermoelastic Analysis of the Response of a Stationary Crack Using the Displacement Discontinuity Method

Thermally induced volumetric changes in rock result in pore pressure variations, and lead to a coupling between the thermal and poromechanical processes. This paper examines the response of a fracture in porothermoelastic rock when subjected to stress, pore pressure, and temperature perturbations. The contribution of each mechanism to the temporal variation of fracture opening is studied to elucidate its effect. This is achieved by development and use of a transient displacement discontinuity (DD) boundary element method for porothermoelasticity. While the full range of the crack opening due to the applied loads is investigated with the porothermoelastic DD, the asymptotic crack opening is ascertained analytically. Good agreement is observed between the numerical and analytical calculations. The results of the study show that, as expected, an applied stress causes the fracture to open while a pore pressure loading reduces the fracture width (aperture). In contrast to the pore pressure effect, cooling of the crack surfaces increases the fracture aperture. It is found that the impact of cooling can be more significant when compared to that of hydraulic loading (i.e., an applied stress and pore pressure) and can cause significant permeability enhancement, particularly for injection/extraction operations that are carried out over a long period of time in geothermal reservoirs.     

Analytical Elastic Solution Based on Fourier Series for a Laterally Confined Granular Column

This paper presents an analytical solution methodology for the complete stress and displacement fields of a laterally confined granular column loaded from the top end. The granular column is idealized as a homogeneous isotropic elastic medium with Coulomb’s friction at the lateral boundary. The solution methodology consists of an analytical procedure that incorporates a potential approach with trigonometric series and Bessel functions, finite Fourier transforms and the superposition method, and an iterative algorithm to satisfy the Coulomb’s friction condition at the lateral boundary. Stress and displacement fields are computed for a specific example and found completely consistent with corresponding finite element results. Key characteristics, computational errors, the convergence behavior, and restrictions of the present approach are discussed. The methodology developed herein can be beneficially applied in the validation process of numerical simulation techniques in granular mechanics such as finite or discrete element methods.     

Well-Balanced Bottom Discontinuities Treatment for High-Order Shallow Water Equations WENO Scheme

A finite volume well-balanced weighted essentially nonoscillatory (WENO) scheme, fourth-order accurate in space and time, for the numerical integration of shallow water equations with the bottom slope source term, is presented. The main novelty introduced in this work is a new method for managing bed discontinuities. This method is based on a suitable reconstruction of the conservative variables at the cell interfaces, coupled with a correction of the numerical flux based on the local conservation of total energy. Further changes regard the treatment of the source term, based on a high-order extension of the divergence form for bed slope source term method, and the application of an analytical inversion of the specific energy-depth relationship. Two ad hoc test cases, consisting of a steady flow over a step and a surge crossing a step, show the effectiveness of the method of treating bottom discontinuities. Several standard one-dimensional test cases are also used to verify the high-order accuracy, the C-property, and the good resolution properties of the resulting scheme, in the cases of both continuous and discontinuous bottoms. Finally, a comparison between the fourth-order scheme proposed here and a well-established second-order scheme emphasizes the improvement achieved using the higher-order approach.     

Visual control of step length during running over irregular terrain.

Examined how visually regulating step length to secure proper footing when running over uneven ground is achieved, studying 2 male experienced long-distance runners. Ss ran on a treadmill on a series of irregularly spaced targets. The movements of their lower limbs and coccyx relative to the targets were monitored opto-electronically by a Selspot system. Results indicate that step length was adjusted to strike the targets primarily by varying the vertical component of impulse applied to the ground during the stance phase. In contrast, horizontal impulse was not varied significantly, and changing the reach forward of the foot on landing contributed little to variation in step length. Changing the vertical impulse simply altered the step time proportionately. Findings are consistent with a time-based model in which vertical impulse is modulated by the optic variable that specifies the time gap that has to be bridged by the runner between 2 targets. (37 ref) (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

Structural Control by Temporal Finite Elements

The shape control of structural systems using the temporal finite element method (TFEM) is addressed in this work. In particular, the standard time marching TFEM is used to predict the displacements of each degree of freedom (DOF) at the end of a given time interval. If these displacements do not exceed the limits imposed on the structure for serviceability, safety, etc., then the displacements for the next time step are computed. If, however, any of the displacements exceed their limit, a control velocity is computed and applied at the beginning of the time interval for each DOF exceeding its limit. The magnitude and direction of this velocity are such as to render the displacement at the end of the time interval equal to the limit value. An example of a two-degree-of-freedom system serves to illustrate this methodology.     

Discrete Particle Distribution Model for Advection-Diffusion Transport

A new methodology, named DisPar, based on a discrete probability distribution for a particle displacement, was developed to solve 1D advection-diffusion transport problems in water bodies. The discrete probability distribution for the particle displacement was developed as an average and variance function. These probabilities were used to predict the deterministic mass transfer between cells in one time step, and therefore the particle concentration in each cell was considered the state variable. The state equation was found to be similar to an explicit finite-difference formulation with a Eulerian grid. The model stability, positivity, and mass conservation are guaranteed by the probability distribution concept. DisPar produces solutions without numerical dispersion for constant velocity, diffusion coefficient, and cross-sectional area. In these conditions, DisPar was also developed as a function of space and time for an instantaneous mass spill. When the stability and positivity restrictions were respected, the model produced excellent results when compared to analytical solutions and other methods. The discrete particle displacement distribution concept differs from other numerical formulations, and therefore it represents a new modeling technique.