New Pseudo Functions To Control Numerical Dispersion

1975 ◽  
Vol 15 (04) ◽  
pp. 269-276 ◽  
Author(s):  
J.R. Kyte ◽  
D.W. Berry
Keyword(s):  
Capillary Pressure ◽  
Cross Section ◽  
Three Dimensional ◽  
Vertical Cross Section ◽  
Numerical Dispersion ◽  
Two Dimensional ◽  
Cross Sectional ◽  
One Dimensional ◽  
Sectional Model ◽  
The Cross

Abstract This paper presents an improved procedure for calculating dynamic pseudo junctions that may be used in two-dimensional, areal reservoir simulations to approximate three-dimensional reservoir behavior. Comparison of one-dimensional areal and two-dimensional vertical cross-sectional results for two example problems shows that the new pseudos accurately transfer problems shows that the new pseudos accurately transfer the effects of vertical variations in reservoir properties, fluid pressures, and saturations from the properties, fluid pressures, and saturations from the cross-sectional model to the areal model. The procedure for calculating dynamic pseudo-relative permeability accounts for differences in computing block lengths between the areal and cross-sectional models. Dynamic pseudo-capillary pressure transfers the effects of pseudo-capillary pressure transfers the effects of different pressure gradients in different layers of the cross-sectional model to the areal model. Introduction Jacks et al. have published procedures for calculating dynamic pseudo-relative permeabilities fro m vertical cross-section model runs. Their procedures for calculating pseudo functions are procedures for calculating pseudo functions are more widely applicable than other published approaches. They demonstrated that, in some cases, the derived pseudo functions could be used to simulate three-dimensional reservoir behavior using two-dimensional areal simulators. For our purposes, an areal simulator is characterized by purposes, an areal simulator is characterized by having only one computing block in the vertical dimension. The objectives of this paper are to present an improved procedure for calculating dynamic pseudo functions, including a dynamic pseudo-capillary pressure, and to demonstrate that the new procedure pressure, and to demonstrate that the new procedure generally is more applicable than any of the previously published approaches. The new pseudos previously published approaches. The new pseudos are similar to those derived by jacks et al. in that they are calculated from two-dimensional, vertical cross-section runs. They differ because (1) they account for differences in computing block lengths between the cross-sectional and areal models, and (2) they transfer the effects of different flow potentials in different layers of the cross-sectional potentials in different layers of the cross-sectional model to the areal model. Differences between cross-sectional and areal model block lengths are sometimes desirable to reduce data handling and computing costs for two-dimensional, areal model runs. For very large reservoirs, even when vertical calculations are eliminated by using pseudo functions, as many as 50,000 computing blocks might be required in the two-dimensional areal model to minimize important errors caused by numerical dispersion. The new pseudos, of course, cannot control numerical pseudos, of course, cannot control numerical dispersion in the cross-sectional runs. This is done by using a sufficiently large number of computing blocks along die length of the cross-section. The new pseudos then insure that no additional dispersion will occur in the areal model, regardless of the areal computing block lengths. Using this approach, the number of computing blocks in the two-dimensional areal model is reduced by a factor equal to the square of the ratio of the block lengths for the cross-sectional and areal models. The new pseudos do not prevent some loss in areal flow-pattern definition when the number of computing blocks in the two-dimensional areal model is reduced. A study of this problem and associated errors is beyond the scope of this paper. Our experience suggests that, for very large reservoirs with flank water injection, 1,000 or 2,000 blocks provide satisfactory definition. Many more blocks provide satisfactory definition. Many more blocks might be required for large reservoirs with much more intricate areal flow patterns. The next section presents comparative results for cross-sectional and one-dimensional areal models. These results demonstrate the reliability of the new pseudo functions and illustrate their advantages pseudo functions and illustrate their advantages over previously derived pseudos for certain situations. The relationship between two-dimensional, vertical cross-sectional and one-dimensional areal reservoir simulators has been published previously and will not be repeated here in any detail. Ideally, the pseudo functions should reproduce two-dimensional, vertical cross-sectional results when they are used in the corresponding one-dimensional areal model. SPEJ P. 269

Long waves in a straight channel with non-uniform cross-section

2015 ◽  
Vol 770 ◽  
pp. 156-188 ◽  
Author(s):  
Patricio Winckler ◽  
Philip L.-F. Liu
Keyword(s):  
Boundary Layer ◽  
Wave Propagation ◽  
Cross Section ◽  
Three Dimensional ◽  
Channel Cross Section ◽  
Cross Sectional ◽  
New Model ◽  
One Dimensional ◽  
Uniform Channel ◽  
The Cross

A cross-sectionally averaged one-dimensional long-wave model is developed. Three-dimensional equations of motion for inviscid and incompressible fluid are first integrated over a channel cross-section. To express the resulting one-dimensional equations in terms of the cross-sectional-averaged longitudinal velocity and spanwise-averaged free-surface elevation, the characteristic depth and width of the channel cross-section are assumed to be smaller than the typical wavelength, resulting in Boussinesq-type equations. Viscous effects are also considered. The new model is, therefore, adequate for describing weakly nonlinear and weakly dispersive wave propagation along a non-uniform channel with arbitrary cross-section. More specifically, the new model has the following new properties: (i) the arbitrary channel cross-section can be asymmetric with respect to the direction of wave propagation, (ii) the channel cross-section can change appreciably within a wavelength, (iii) the effects of viscosity inside the bottom boundary layer can be considered, and (iv) the three-dimensional flow features can be recovered from the perturbation solutions. Analytical and numerical examples for uniform channels, channels where the cross-sectional geometry changes slowly and channels where the depth and width variation is appreciable within the wavelength scale are discussed to illustrate the validity and capability of the present model. With the consideration of viscous boundary layer effects, the present theory agrees reasonably well with experimental results presented by Chang et al. (J. Fluid Mech., vol. 95, 1979, pp. 401–414) for converging/diverging channels and those of Liu et al. (Coast. Engng, vol. 53, 2006, pp. 181–190) for a uniform channel with a sloping beach. The numerical results for a solitary wave propagating in a channel where the width variation is appreciable within a wavelength are discussed.

Pseudofunctions for Water Coning in a Three-Dimensional Reservoir Simulator

1977 ◽  
Vol 17 (04) ◽  
pp. 251-262 ◽  
Author(s):  
E.G. Woods ◽  
A.K. Khurana
Keyword(s):  
Capillary Pressure ◽  
Cross Section ◽  
Three Dimensional ◽  
Production Performance ◽  
Vertical Cross Section ◽  
Two Dimensional ◽  
Well Performance ◽  
Reservoir Model ◽  
Reservoir Simulator ◽  
Static Conditions

Abstract Three-dimensional numerical models of bottom-water-drive reservoirs show delayed water breakthrough into individual wells when compared with observed well performance and individual-well coning models. This reservoir-model behavior results from masking of the well coning effect by volume-averaging pressure and saturation profiles around a well over a grid block with a large volume. The reservoir-simulator prediction of well performance can be improved by mathematically performance can be improved by mathematically transforming the production performance of a detailed well-coning model into a set of time-independent pseudorelative-permeability and capillary-pressure curves that then can be used in the reservoir model. A procedure for obtaining the required pseudofunctions is described and the results of their application in simple models and in a large reservoir-simulator model are shown. Introduction The prohibitive cost of numerical reservoir simulation with fine-grid definition models of large reservoirs has led to development of techniques whereby vertical saturation distribution and/or localized flow conditions in the vicinity of individual wells can be approximately accounted for in relatively coarse-grid models at an acceptable incremental cost. In particular, vertical cross-section models under capillary and gravity equilibrium have been used to derive pseudorelative permeabilities and capillary pressures for use in two-dimensional, areal models to simulate the average vertical distribution of flow without having to pay the computing price of a full three-dimensional model. Coats et al. described the use of the vertical equilibrium concept for developing pseudorelative-permeability and capillary-pressure pseudorelative-permeability and capillary-pressure functions for simulating the vertical dimension in a two-dimensional, areal simulator model This method assumes gravity-capillary equilibrium in the vertical direction. Also, Coats et al. developed a dimensionless parameter for estimating when these conditions are valid. Martin formed a mathematical basis for pseudofunctions by reducing the equations for pseudofunctions by reducing the equations for three-phase, three-dimensional, compressible flow to two-dimensional relations by partial integration of the equations of flow. Hearn extended the pseudorelative-permeability concept by adapting it pseudorelative-permeability concept by adapting it to stratified reservoirs where viscous rather than gravity and capillary forces dominate the vertical sweep efficiency. Hawthorne studied the effects of capillary pressure on pseudorelative permeability derived from the Hearn stratified model. Jacks et al. further enlarged thepseudorelative-perrneability concept by developing dynamic pseudorelative permeabilities. (Dynamic pseudos, denoting pseudos permeabilities. (Dynamic pseudos, denoting pseudos determined under flowing rather than static conditions, allow one to account for the interaction between viscous and gravity forces resulting from rate variation in the vertical plane.) Kyte and Berry generalized the work of Jacks et al. by introducing the concept of pseudocapillary pressures and improving dynamic pseudofunction calculations to include varying flow potential gradients. Emanual and Cook expanded the concept of vertical cross-section, pseudorelative permeabilities to include the vertical performance of individual wells. Their procedure combines the effect of coning and well pseudorelative permeabilities for use in a two-dimensional, areal model. Chappelear and Hirasaki used a different approach to including of coning effects in a two-dimensional, areal simulator by developing a functional relationship among water cut, average oil-column thickness, and total rate based on an analytical incompressible, steady-state model. The most sophisticated approach to representing well-coning effects in a reservoir simulator has been taken by Mrosovsky and Ridings and Akbar et al. They incorporated detailed numerical well models into the reservoir-model grid. SPEJ P. 251

scholarly journals Hierarchical Beam Finite Elements Based Upon a Variables Separation Method

2016 ◽  
Vol 08 (02) ◽  
pp. 1650026 ◽  
Author(s):  
Gaetano Giunta ◽  
Salim Belouettar ◽  
Olivier Polit ◽  
Laurent Gallimard ◽  
Philippe Vidal ◽  
...  
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Cross Section ◽  
Three Dimensional ◽  
Finite Element Solution ◽  
Stress Fields ◽  
Element Solution ◽  
One Dimensional ◽  
Kinematic Approximation ◽  
The Cross

A family of hierarchical one-dimensional beam finite elements developed within a variables separation framework is presented. A Proper Generalized Decomposition (PGD) is used to divide the global three-dimensional problem into two coupled ones: one defined on the cross-section space (beam modeling kinematic approximation) and one belonging to the axis space (finite element solution). The displacements over the cross-section are approximated via a Unified Formulation (UF). A Lagrangian approximation is used along the beam axis. The resulting problems size is smaller than that of the classical equivalent finite element solution. The approach is, then, particularly attractive for higher-order beam models and refined axial meshes. The numerical investigations show that the proposed method yields accurate yet computationally affordable three-dimensional displacement and stress fields solutions.

The Modeling of a Three-Dimensional Reservoir with a Two-Dimensional Reservoir Simulator-The Use of Dynamic Pseudo Functions

1973 ◽  
Vol 13 (03) ◽  
pp. 175-185 ◽  
Author(s):  
Hugh H. Jacks ◽  
Owen J.E. Smith ◽  
C.C. Mattax
Keyword(s):  
Capillary Pressure ◽  
Cross Section ◽  
Relative Permeability ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Relative Permeabilities ◽  
Large Reservoir ◽  
Model Simulations ◽  
Wide Range ◽  
Section Model

Abstract Dynamic pseudo-relative permeabilities derived from cross-section models can be used to simulate three-dimensional flow accurately in a two-dimensional areal model of a reservoir Techniques are presented for deriving and using dynamic pseudos that are applicable over a wide range of rates and initial fluid saturations. Their validity is demonstrated by showing calculated results from comparable runs in a vertical cross-section model and in a one-dimensional areal model using the dynamic pseudo-relative permeabilities and vertical equilibrium (VE) pseudo-capillary pressures. Further substantiation is provided by showing the close agreement in calculated performance for a three-dimensional model and corresponding two-dimensional areal model representing a typical pattern on the flanks of a large reservoir. The areal pattern on the flanks of a large reservoir. The areal model gave comparable accuracy with a substantial savings in computing and manpower costs. Introduction Meaningful studies can be made for almost all reservoirs now that relatively efficient three-dimensional reservoir simulators are available. In many instances, however, less expensive two-dimensional areal (x-y) models can be used to solve the engineering problem adequately, provided the nonuniform distribution and flow of fluids in the implied third, or vertical, dimension of the areal model is properly described. This is accomplished through the use of special saturation-dependent functions that have been labeled pseudo-relative permeability (k ) and pseudo-capillary pressure permeability (k ) and pseudo-capillary pressure (P ) or, for simplicity "pseudo functions", to distinguish them from the conventional laboratory measured values that are used in their derivation. Two types of reservoir models have been suggested in the past to derive pseudo functions: the vertical equilibrium (VE) model of Coats et al., which is based on gravity-capillary equilibrium in the vertical direction; and the stratified model of Hearn, which assumes that viscous forces dominate vertical fluid distribution. Neither of these models accounts for the effects of large changes in flow rate that take place as a field is developed, approaches and place as a field is developed, approaches and maintains its peak rate, and then falls into decline. This paper presents an alternative method for developing pseudo functions that are applicable over a wide range of flow rates and over the complete range of initial fluid saturations. The functions may be both space and time dependent and, again for clarity and convenience in nomenclature, we have labeled them "dynamic pseudo functions". DESCRIPTION OF PSEUDO-RELATIVE PERMEABILITY FUNCTIONS PERMEABILITY FUNCTIONS Methods for developing pseudo functions have been presented in the literature. The distinction between our method and those used by others lies in the technique for deriving the vertical saturation distribution upon which the pseudo-relative permeabilities are based. In our approach, the permeabilities are based. In our approach, the vertical saturation distribution is developed through detailed simulation of the fluid displacement in a vertical cross-section (x-z) model of the reservoir. The simulation is run under conditions that are representative of those to be expected during the period to be covered in the areal model simulations. period to be covered in the areal model simulations. Results of the cross-section simulation are then processed to give depth-averaged fluid saturations processed to give depth-averaged fluid saturations (S ) and "dynamic" pseudo-relative permeability values (k ) for each column of blocks in the cross-section model at each output time. The above approach can result in a different set of dynamic pseudo functions for each column of blocks due to differences in initial saturation, rate of displacement, reservoir stratification, and location. However, differences between columns are frequently minor or they can be accounted for by correlation of the data. In this and several other reservoir studies, it was possible to reduce the complexity of the pseudo function sets through correlations with initial fluid saturations and fluid velocities. SPEJ P. 175

Numerical Three-Phase Model of the Two-Dimensional Steamflood Process

1970 ◽  
Vol 10 (04) ◽  
pp. 405-417 ◽  
Author(s):  
N.D. Shutler
Keyword(s):  
Fluid Flow ◽  
Cross Section ◽  
Oil Recovery ◽  
Vertical Cross Section ◽  
Recovery Efficiency ◽  
Two Dimensional ◽  
Dimensional Model ◽  
One Dimensional ◽  
Oil Water ◽  
Three Phases

Abstract This paper describes a numerical mathematical model that is a significant extension of a previously published one-dimensional model of the steamflood published one-dimensional model of the steamflood process. process. The model describes the simultaneous flow of the three phases - oil, water and gas - in two dimensions. Interphase mass transfer between water and gas phases is allowed, but the oil is assumed nonvolatile and the hydrocarbon gas insoluble in the liquid phases. The model allows two-dimensional heat convection within the reservoir and two-dimensional heat conduction in a vertical cross-section spanning the oil sand and adjacent strata. Example calculations are presented which, on comparison with experimental results, tend to validate the model. Steam overriding due to gravity effects is shown to significantly reduce oil recovery efficiency in a thick system while jailing to do so in a thinner system. A study of the effect of capillary pressure indicates that failure to scale capillary forces in laboratory models of thick sands may lead to optimistic recovery predictions, while properly scaled capillary forces may be sufficiently low as to play no important role in oil recovery. Calculations made with and without vertical permeability show that failure to account for vertical fluid flow can lead to predictions of pessimistic oil recovery efficiency. pessimistic oil recovery efficiency Introduction Mathematical tools of varying complexity have been used in studying the steamflood process. A "simplified" class of mathematical models has served primarily as aids in engineering design. A more comprehensive class of models has improved understanding of the nature of the process. The model described in this report is of the latter class, but it is more comprehensive than any previously published model. published model. All previously available calculations of the steamflood process are confined to one space dimension in their treatments of fluid flow. Thus all previous models necessary ignore all effects of gravity reservoir heterogeneity, and nonuniform initial fluid-phase distributions on fluid flow in a second dimension. This model, an extension of a previously published model accounts for heat and previously published model accounts for heat and fluid transfer in two space dimensions and, hence, can evaluate these effects on simultaneous horizontal and vertical flow. While the model can describe the areal performance of a steamflood (in which case the heat transfer is described in three dimensions), this aspect will not be considered in this paper. Rather, this paper will describe the model in its application to a vertical cross-section through the reservoir and will consider some preliminary investigations to demonstrate the importance of being able to simultaneously account for horizontal and vertical fluid flow. Mathematical details are given in appendices. MATHEMATICAL DESCRIPTION OF STEAMFLOODING Darcy's law provides expressions for the velocities of the three phases (oil, water and gas), which, when combined with oil, water and gas mass balances give the partial differential equations governing Now of the three phases within a reservoir sand: OIL PHASE ..(1) WATER PHASE ..(2) SPEJ P. 405

Sound Attenuation and Prediction of Porous Media Properties in Hybrid Ducts Utilizing Spatially Periodic Area Changes

2008 ◽  
Author(s):  
Lisa J. Burton ◽  
Donald B. Bliss ◽  
Linda P. Franzoni
Keyword(s):  
Porous Material ◽  
Cross Section ◽  
Pass Band ◽  
Cross Sectional ◽  
Open Cell ◽  
One Dimensional ◽  
Open Cell Foam ◽  
Spatially Periodic ◽  
The Cross ◽  
Cell Foam

A theory based on cross-sectional averaging is developed to analyze quasi-one-dimensional acoustic propagation in hybrid ducts with two propagation media in the cross-section. Specifically, ducts lined with a thick layer of porous material are considered. The porous material makes the duct wavenumber complex, changing the phase speed and introducing attenuation. To lowest order, the wavenumber depends only on the ratio of cross-sectional areas and the properties of the constituent media, and surprisingly not on the material configuration in the cross-section. High frequency accuracy can be improved by using a small correction that includes shape coefficients that depend on the cross-sectional configurations. If the propagation wavenumber is measured experimentally in a hybrid duct, the complex effective sound speed and density, fundamental porous material properties, can be extracted relatively easily. Experimentally, open cell foam samples line the sides of a tube closed at one end, and the complex wavenumber is determined from standing wave measurements. The cross-sectional averaging theory is then used to determine the acoustic properties of the open-cell foam. Results are compared for various lining configurations to assess the accuracy of the method. Another application of this work is the theoretical and experimental study of the propagation of quasi one-dimensional acoustic waves through a duct with spatially periodic area changes. This configuration exhibits stop-band and pass-band behavior, with substantially reduced sound transmission in stop bands, but little effect in pass bands. The regions of the duct with larger cross-sectional area are partially filled with an annular region of porous material to provide pass-band attenuation, leaving a constant area passage for airflow. Predictions and measurements for hybrid ducts with periodic area changes are presented. A muffler designed to place engine harmonics in targeted stop-bands is described.

An Experimental Verification of a Two-Dimensional Technique for Computing Performance Of Gas-Drive Reservoirs

1963 ◽  
Vol 3 (01) ◽  
pp. 19-27 ◽  
Author(s):  
P.M. Blair ◽  
D.W. Peaceman
Keyword(s):  
Cross Section ◽  
Space Dimension ◽  
Vertical Cross Section ◽  
Low Flow ◽  
Two Dimensional ◽  
Gas Oil ◽  
One Dimensional ◽  
Flow Equations ◽  
The One ◽  
One Space Dimension

Abstract The shape and position of the gas-oil transition zone during downdip displacement of oil by gas has been calculated using flow equations which include the effects of gravity, relative permeability, capillary pressure and compressibility of the fluids. The calculations treat the problem in two space dimensions, and results are compared with data from a laboratory model tilted at 30 degrees and 60 degrees from the horizontal on displacements near and above the maximum rate at which gravity segregation prevents channeling of the gas along the top of the stratum. The good agreement between calculated and experimental results demonstrates the validity of the technique as well as that of the flow equations. Introduction Knowledge of the fluid distribution and movement in and oil reservoirs important in producing operations and estimation of reserves. The history of the oil industry has included steady progress in improving the accuracy of calculations which provide the required knowledge. The earliest method of calculating reservoir performance consisted of material-balance equations based on the assumption that all properties were uniform throughout a reservoir. For many reservoirs such a simple formulation is still the most useful. However, when large pressure and saturation gradients exist in a reservoir, the assumption of uniform values throughout may lead to significant error. To reduce these errors, Buckley and Leverett introduced a displacement equation which considers pressure and saturation gradients. Methods available at that time permitted solutions to the Buckley-Leverett equation in one space dimension; these solutions have been very useful in solving many problems related to the production of oil. However, the one-dimensional methods are not adequate for systems in which saturations vary in directions other than the direction of flow. An example of such a system is the case of gas displacing oil down a dipping stratum in which the gas-oil contact becomes significantly tilted. Of course, the Buckley-Leverett displacement method cannot predict the tilt of the gas-oil contact. Recent improvements of the one-dimensional Buckley-Leverett method achieve some success in predicting the tilt of the gas-oil contact at sufficiently low flow rates. However, at rates high enough that the viscous pressure gradient nearly equals or exceeds the gravity gradient, even these improved one-dimensional methods incorrectly predict the shape and velocity of the contact. Further progress in estimating such fluid movements in a reservoir appears to require consideration of the problem in more than one space dimension. The recent two-dimensional method of Douglas, Peaceman and Rachford appears adaptable to calculate changes with time of the saturation distribution in a vertical cross-section of a reservoir. The movement of saturation contours should represent the moving fluid contacts and include the effects of crossflow due to gravity, as well as variations in the rock and fluid properties. The nonlinear nature of the equations used in the method has prevented proof of the validity of the solutions. Douglas, Peaceman and Rachford made some comparisons with experiment but did not include cases in which gravity was important nor cases involving displacement by the nonwetting phase. Forthesereasons, atestof the two-dimensional method for a case in which these factors are included would be very desirable. The test selected was a comparison of calculated results with those from a carefully controlled laboratory experiment on a model with measured physical properties. The model selected was one in which gas displaced oil down a tilted, rectangular sand pack. The model can be thought of as representing a vertical cross-section taken parallel to the dip of a reservoir. The displacement thus simulates gas displacing oil downdip that might result from gas-cap expansion or gas injection. SPEJ P. 19^

Investigation of Cell-Sensor Hybrid Structures by Focused Ion Beam (FIB) Technology

2006 ◽  
Vol 983 ◽  
Author(s):  
Andreas Heilmann ◽  
Frank Altmann ◽  
Andreas Cismak ◽  
Werner Baumann ◽  
Mirko Lehmann
Keyword(s):  
Cross Section ◽  
Mammalian Cells ◽  
Focused Ion Beam ◽  
Ion Beam ◽  
Three Dimensional ◽  
Nerve Cells ◽  
Cross Sectional ◽  
Dimensional Reconstruction ◽  
The Cross ◽  
Silicon Interface

AbstractFor the investigation of the adhesion of mammalian cells on a semiconductor biosensor structure, nerve cells on silicon neurochips were prepared for scanning electron microscopy investigations (SEM) and cross-sectional preparation by focused ion beam technology (FIB). The cross-sectional pattern demonstrates the focal adhesion points of the nerve cells on the chip. Finally, SEM micrographs were taken parallel to the FIB ablation to investigate the cross section of the cells slice by slice in order to demonstrate the spatial distribution of focal contact positions for a possible three-dimensional reconstruction of the cell-silicon interface.

Kinetics-Induced Morphing of Three-Dimensional-Printed Gel Structures Based on Geometric Asymmetry

2020 ◽  
Vol 87 (7) ◽  
Author(s):  
Qi Li ◽  
Zhao Xu ◽  
Suchun Ji ◽  
Pengyu Lv ◽  
Xiying Li ◽  
...  
Keyword(s):  
Cross Section ◽  
Polymer Networks ◽  
Three Dimensional ◽  
Precursor Solution ◽  
Dynamic Responses ◽  
Cross Sectional ◽  
Polymeric Gels ◽  
The Cross ◽  
External Stimuli ◽  
Printing Parameters

Abstract Emerging three-dimensional (3D) printing techniques for soft active materials have demonstrated fascinating applications in various areas including programmable and reconfigurable structures, tissue engineering, and soft robotics. For example, polymeric gels, which consist of polymer networks swollen with solvent molecules, are capable of deforming and swelling/deswelling in response to external stimuli. Although polymeric gels are used to print structures, little attention has been paid to the effect of printing parameters on the cross-sectional shape of 3D-printed gel filaments or further to the dynamic responses of the printed structures. Due to the flow of the precursor solution before fully cured, the cross section of a printed gel filament is usually asymmetric. When immersed in water, the asymmetry in the cross section causes the printed filament to bend, and the interdiffusion of the two solvents leads to the alternation in bending direction. The bending curvature and response rate can be adjusted by turning printing parameters. As applications of this mechanism, we demonstrated various types of gel structures, capable of deforming from 1D strips to 2D spiral or sinusoidal shapes, warping from 2D flat sheet to 3D cylindrical helix when swollen, or wrapping and manipulating objects under external stimuli.

NONLINEAR THIN-WALLED BEAMS WITH A RECTANGULAR CROSS-SECTION — PART I

2012 ◽  
Vol 22 (03) ◽  
pp. 1150016 ◽  
Author(s):  
LORENZO FREDDI ◽  
MARIA GIOVANNA MORA ◽  
ROBERTO PARONI
Keyword(s):  
Cross Section ◽  
Elastic Energy ◽  
Rectangular Cross Section ◽  
Three Dimensional ◽  
Scaling Factor ◽  
One Dimensional ◽  
Thin Walled ◽  
The Cross ◽  
Limit Models ◽  
Cross Section Part

Our aim is to rigorously derive a hierarchy of one-dimensional models for thin-walled beams with rectangular cross-section, starting from three-dimensional nonlinear elasticity. The different limit models are distinguished by the different scaling of the elastic energy and of the ratio between the sides of the cross-section. In this paper we report the first part of our results. More precisely, denoting by h and δh the length of the sides of the cross-section, with δh ≪ h, and by [Formula: see text] the scaling factor of the bulk elastic energy, we analyze the cases in which δh/εh → 0 (subcritical) and δh/εh → 1 (critical).

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