Superposition Formulation Research Articles

Nonlinear viscoelastic properties of human dentin under uniaxial tension

ObjectiveDentin is a viscoelastic tissue that contributes to the load distribution in human teeth and leads to their fracture resistance. Despite previous researches on the time-dependent behavior of dentin, it is not very clear whether the viscoelastic behavior of this tissue is linear or nonlinear, and what viscoelastic constitutive equations mechanically characterize it. Therefore, the aim of this study was to describe the viscoelastic behavior of human dentin and determine the best-fitting viscoelastic model for this tissue. MethodsAfter preparation of human dentin specimens from 50 subjects, tensile stress relaxation tests were performed at 1%, 3%, 5% and 7% strain amplitudes. We first evaluated the viscoelastic linearity of this tissue and then fitted the experimental data using different constitutive models, namely, 2-, 3- and 4-term Prony series for linear viscoelasticity, Fung’s quasilinear viscoelastic model, and also Schapery and modified superposition models for nonlinear viscoelasticity. ResultsDespite an almost linear trend at small strains up to 5%, the relaxation rate generally depended on strain amplitude, indicating some degree of nonlinearity in dentine viscoelasticity. According to the results of data fitting using different models, the modified superposition formulation could best capture the viscoelastic behavior of human dentin. SignificanceIn this study, we have quantitatively examined the viscoelastic behavior of human dentin, using a large number of samples. We have obtained the coefficients of various viscoelastic formulations, which can be utilized in subsequent researches on human dentin assuming linear, quasilinear or nonlinear viscoelasticity for this tissue.

A Simple Superposition Formulation to Predict the Underwater Electric Potential Signature of Naval Vessels

The underwater electric potential (UEP) signature is an electric signal, which can be exploited by naval mines to be utilized as a possible trigger indicator and may cause severe damage to the vessel and the onboard crew. Hence, knowing the UEP signature as exactly as possible can help to evaluate a possible risk of the vessel being detected by naval mines or if the UEP signature is within a noncritical region. As the UEP signature differs for changes of the corrosion protection system, the UEP signature is usually unknown for new conditions. In this work, we present a simple mathematical formulation to predict the UEP signature based on the mere use of a single reference UEP signature, and the corresponding currents, which are excited by the impressed current cathodic protection (ICCP) system. With this methodology, deviations below 10% between the maximum of the simulated UEP signature and the predicted UEP signature can be achieved, even in the presence of the nonlinear corrosion process. Furthermore, a corrosion protective coating of the propellers can significantly reduce the influence of the nonlinear corrosion process on the total UEP signature to improve the prediction accuracy of the superposition formulation as presented in this work.

An analytical solution of fractional burgers equation

Using the fractional complex transform, the fractional partial differential equations can be reduced to ordinary differential equations which can be solved by the auxiliary equation method. Non-linear superposition formulation of Riccati equation is applied, and a complex infinite sequence solution is obtained.

Coupled MPS-modal superposition method for 2D nonlinear fluid-structure interaction problems with free surface

In this paper, a coupled MPS-modal superposition method is developed for 2D nonlinear fluid-structure interaction problems. In this method, the rigid-body and relatively small elastic deformation are coupled together, which considers the mutual effect between them. The elastic deformation of the structure is represented by a mode superposition formulation, which is more efficient compared with FEM, regardless of the size of the structure. For 2D cases, if the first three modes are chosen to represent the flexible deformation of the structure, it only results in a 6×6 matrix equation to be solved. For the fluid motion, the modified Moving Particle Semi-implicit (MPS) method, which significantly reduces the fluctuation of pressure calculation of the original MPS method, is used.Two nonlinear problems, i.e. breaking-water-dam impacting a floating beam and flexible wedge slamming into the water are simulated to demonstrate the performance of the developed method. The numerical simulations show that this coupling model is capable of providing stable results that are generally in good agreement with the available experimental data. For the highly nonlinear case with very large rigid motions, the mutual effect between elastic deformation and rigid motions could accumulate to a relatively remarkable level shown by the curves of trajectories or acceleration history of the body mass centre. This also indicates the importance of mutual effect to analyse highly nonlinear FSI problems with large rigid-body motions and relatively small flexible deformation.

Integrated Field-Scale Production and Economic Evaluation Under Subsurface Uncertainty for the Pattern-Driven Development of Unconventional Resources With Analytical Superposition

SummaryThe in-situ upgrading process (IUP) is a thermal-recovery technique that relies on a pattern-based development process, a complicated physical process that involves thermal and mass transfer in porous media, which renders full field-scale reservoir simulations impractical. Although it is feasible to quantify the impact of subsurface uncertainties on recovery for small-scale sector models with experimental design (ED), it is still a very challenging problem to quantify their impact on field-scale quantities. Straightforward upscaling to field scale does not work because such conventional superposition-based methods do not capture the effects of spatial variability in rock and fluid properties and the time delay in sequential pattern development.In this paper, we show that, under certain mild assumptions, an analytical superposition formulation can be developed that propagates the uncertainties of production forecasts and economic evaluations generated from a sector model to full field-scale quantities. One can simplify this formulation further so that the variance of a field-scale quantity is analytically expressed as the variance of the same single-pattern quantity multiplied by a (computable) scaleup factor. This makes it possible to implement a practical uncertainty quantification work flow in which single-pattern results are upscaled to accurate full field results with reliable uncertainty ranges, without the need for full field-scale simulations.We apply the proposed novel superposition and uncertainty-propagation method to a multipattern IUP development, and demonstrate that this work flow produces reliable results for field-scale production and economics as well as realistic uncertainty ranges. Moreover, these results indicate that the scaleup factor for single-pattern results can accurately capture the impact of spatial correlations of subsurface uncertainties, the size of the field-scale model, the time-delay in pattern development, and the discount rate.Uncertainty quantification of field-scale production and economics is a key factor for the successful development of unconventional resources such as extraheavy oil and oil shale with significant rewards in terms of risk management and project profitability. With minor modifications, the proposed method can also be applied to other pattern-driven processes such as the in-situ conversion process (ICP) and steam-assisted gravity drainage.

An Efficient Optimization Work Flow for Field-Scale In-Situ Upgrading Developments

Summary In-situ upgrading process (IUP) is an attractive technology for developing unconventional extraheavy-oil reserves. Decisions are generally made on field-scale economics evaluated with dedicated commercial tools. However, it is difficult to conduct an automated IUP optimization process because of unavailable interface between the economic evaluator and commercial simulator/optimizer, and because IUP is such a highly complex process that full-field simulations are generally not feasible. In this paper, we developed an efficient optimization work flow by addressing three technical challenges for field-scale IUP developments. The first challenge was deriving an upscaling factor modeled after analytical superposition formulation; proposing an effective method of scaling up simulation results and economic terms generated from a single-pattern IUP reservoir-simulation model to field scale; and validating this approach numerically. The second challenge was proposing a response-surface model (RSM) of field economics to analytically compute key field economical indicators, such as net present value (NPV), by use of only a few single-pattern economic terms together with the upscaling factor, and validating this approach with a commercial tool. The proposed RSM approach is more efficient, accurate, and convenient because it requires only 15–20 simulation cases as training data, compared with thousands of simulation runs required by conventional methods. The third challenge is developing a new optimization method with many attractive features: well-parallelized, highly efficient and robust, and with a much-wider spectrum of applications than gradient-based or derivative-free methods, applicable to problems without any derivative, with derivatives available for some variables, or with derivatives available for all variables. This work flow allows us to perform automated field IUP optimizations by maximizing a full-field economics target while honoring all field-level facility constraints effectively. We have applied the work flow to optimize the IUP development of a carbonate heavy-oil asset. Our results show that the approach is robust and efficient, and leads to development options with a significantly improved field-scale NPV. This work flow can also be applied to other kinds of pattern-based field developments of shale gas and oil, and thermal processes such as steamdrive or steam-assisted gravity drainage.

Nonlinear viscoelastic characterization of the porcine spinal cord

Although quasi-static and quasi-linear viscoelastic properties of the spinal cord have been reported previously, there are no published studies that have investigated the fully (strain-dependent) nonlinear viscoelastic properties of the spinal cord. In this study, stress relaxation experiments and dynamic cycling were performed on six fresh porcine lumbar cord specimens to examine their viscoelastic mechanical properties. The stress relaxation data were fitted to a modified superposition formulation and a novel finite ramp time correction technique was applied. The parameters obtained from this fitting methodology were used to predict the average dynamic cyclic viscoelastic behavior of the porcine cord. The data indicate that the porcine spinal cord exhibited fully nonlinear viscoelastic behavior. The average weighted root mean squared error for a Heaviside ramp fit was 2.8kPa, which was significantly greater (p<0.001) than that of the nonlinear (comprehensive viscoelastic characterization method) fit (0.365kPa). Further, the nonlinear mechanical parameters obtained were able to accurately predict the dynamic behavior, thus exemplifying the reliability of the obtained nonlinear parameters. These parameters will be important for future studies investigating various damage mechanisms of the spinal cord and studies developing high-resolution finite elements models of the spine.

Experimental Characterization and Finite Element Implementation of Soft Tissue Nonlinear Viscoelasticity

Finite element (FE) models of articular joint structures do not typically implement the fully nonlinear viscoelastic behavior of the soft connective tissue components. Instead, contemporary whole joint FE models usually represent the transient soft tissue behavior with significantly simplified formulations that are computationally tractable. The resultant fidelity of these models is greatly compromised with respect to predictions under temporally varying static and dynamic loading regimes. In addition, models based upon experimentally derived nonlinear viscoelastic coefficients that do not account for the transient behavior during the loading event(s) may further reduce the model's predictive accuracy. The current study provides the derivation and validation of a novel, phenomenological nonlinear viscoelastic formulation (based on the single integral nonlinear superposition formulation) that can be directly inputted into FE algorithms. This formulation and an accompanying experimental characterization technique, which incorporates relaxation manifested during the loading period of stress relaxation experiments, is compared to a previously published characterization method and validated against an independent analytical model. The results demonstrated that the static and dynamic FE approximations are in good agreement with the analytical solution. Additionally, the predictive accuracy of these approximations was observed to be highly dependent upon the experimental characterization technique. It is expected that implementation of the novel, computationally tractable nonlinear viscoelastic formulation and associated experimental characterization technique presented in the current study will greatly improve the predictive accuracy of the individual connective tissue components for whole joint FE simulations subjected to static and dynamic loading regimes.

A TUTORIAL PRESENTATION OF ALTERNATIVE SOLUTIONS TO THE FLEXIBLE BEAM ON RIGID CART PROBLEM

A classical planar problem in forward flexible multibody dynamics is thoroughly investigated. The system consists of a damped flexible beam cantilevered to a rigid translating cart. The problem is solved using three distinctly different conventional approaches presented in roughly the chronological order in which they have been applied to flexible dynamic systems. First, a modal superposition formulation based on Bernoulli-Euler beam theory is developed. Second, an alternative solution is developed drawing exclusively on methods for rigid body dynamics combined with a knowledge of the theoretical modal behaviour of continuous beams. Third, a formulation based on the conventional finite element method using four-degree-of-freedom planar beam elements is adapted to include the rigid body motion of the cart. The relative merits of the three formulations are discussed and numerical simulation results generated using each of the three formulations are compared with each other and with a solution from a general-purpose flexible multibody dynamics formulation that is briefly outlined. The relative accuracy and efficiency of the methods and the challenges associated with generalizing each formulation are discussed.

A robust technique to reconstruct sound field for all wave-numbers

A robust technique has been proposed for reconstructing sound field for all wave-numbers, which is suitable for the radiation analysis on an arbitrarily-shaped body. According to the wave superposition formulation, the spatial sound field of an actual radiator can be represented equivalently by superposing the fields generated by some fictitious sources interior to the actual radiator. Meanwhile, the strength density on the fictitious sources can be obtained by matching the sound pressures of a finite number of field points. Once the strength density is calculated, the acoustic information at any field point can be obtained subsequently. Finally, some typical numerical examples are given to validate the theoretical analysis, and some investigations are made to analyze the influence of sound frequency and the fictitious source position on the reconstruction precision, which are helpful to the application of the proposed technique to real industrial measurements.

Electrical Double-Layer Interaction between Heterogeneously Charged Colloidal Particles: A Superposition Formulation

The superposition formula for the electrical double-layer interaction between two uniformly charged spherical particles immersed in an unbounded electrolyte has been used extensively since its formulation some 50 years ago. However, the corresponding result for two heterogeneously charged spherical particles has to date remained elusive. In this paper, we present such analytical results and also consider the special case of a heterogeneously charged spherical particle interacting with a uniformly charged plate. In the sphere–plate case we suggest a simple modification to the superposition result, based on the incorporation of image effects, that can dramatically improve its regime of validity. We assess the validity and accuracy of the superposition formulae by comparison with results of a rigorous boundary element calculation of the electrostatic interaction.

Calculation of total cross sections of multiple-sphere clusters

A method for calculating the extinction, absorption, and scattering cross sections of clusters of neighboring spheres for both fixed and random orientations is developed. The analysis employs the superposition formulation for radiative interactions among spheres, in which the total field from the cluster is expressed as a superposition of vector spherical harmonic expansions about each of the spheres in the cluster. Through the use of addition theorems a matrix equation for the expansion coefficients is obtained. Further application of addition theorems on the inverse of the coefficient matrix is shown to yield analytical expressions for the orientation-averaged total cross sections of the sphere cluster. Calculations of the cross sections of pairs of spheres and fractal aggregates of several spheres are presented. It is found that a dipole representation of the field in each sphere does not adequately predict the absorption cross section of clusters of small-size-parameter spheres when the spheres are highly conducting. For this situation several multipole orders are required for an accurate calculation of the absorption cross section. In addition, the predicted absorption of sphere clusters can be significantly greater than that estimated from the sum of the isolated-sphere cross sections.

Active control of the acoustic radiation of a vibrating structure using a superposition formulation

In this paper, optimal control theory is applied to the problem of reducing the acoustic power radiated from a complex, three-dimensional, vibrating structure. Using the principle of wave superposition, a vibrating structure’s acoustic field can be shown to be approximately equivalent to the sum of the fields due to a finite number of acoustic sources located in the interior of the radiating body. Based on this transformation, the total time-averaged, acoustic power output from the vibrating structure can be expressed in terms of the strengths of the interior point sources. Two methods are then used to optimally reduce the acoustic power output from the vibrating body. The first method consists of placing a secondary source outside the structure and, by optimizing its source strength, minimizing the total, time-averaged acoustic power output of the entire system. The second method assumes that the vibrating structure’s normal surface velocity can be altered such that the total power radiated from the vibrating structure is minimized. The optimum surface velocity distribution produced by this method is shown to be dependent on the number and location of the surface nodes defining the geometry of the structure.

Active control for the acoustic radiation of a vibrating structure using a superposition formulation

A new, computational approach provides a methodology for specifying how the acoustic power radiated from a complex, three-dimensional vibrating structure may be controlled. The method uses a superposition principle to transform the surface of a vibrating structure into a discrete distribution of acoustic sources interior to the boundary surface of the structure. The transformation is carried out by requiring that the interior sources produce the same or nearly the same normal velocities as those prescribed on the surface of the vibrating structure. Based on this transformation, the total time-averaged, acoustic power output from the vibrating structure can be expressed in terms of the strengths of the equivalent sources. To control the radiated acoustic power, one or more secondary point sources is then placed on or near the surface of the vibrating structure. By using the total power radiated from the vibrating structure and the secondary sources as an objective function for an optimization operation, the optimal strengths and locations of the secondary sources are found that lead to a maximum reduction in the acoustic power radiated from the vibrating structure. The significance of this approach is that it provides design guidelines for applying active control techniques to complex vibrating structures.