Applied Mathematical Modelling Research Articles

A method of lines approach for modelling saltwater intrusion in coastal aquifers

The equations governing saltwater intrusion in coastal aquifers are complex. Backward Euler time stepping approaches are often used to advance the solution to these equations in time, which typically requires that small time steps be taken in order to ensure that an accurate solution is obtained. We show that a method of lines approach incorporating variable order backward differentiation formulas can greatly improve the efficiency of the time stepping process. References K. Bajracharya, J. Arunakumaren, and W. J. Huxley. Numerical modelling of seawater intrusion in Gooburrum, Bundaberg, Queensland. In T. R. Weaver and C. R. Lawrence, editors, Proceedings, Groundwater: Sustainable Solutions, pages 613ñ--618. University of Melbourne, 1998. J. Bear. Hydraulics of groundwater. McGraw-Hill, New York, 1979. K. E. Brenan, S. L. Campbell, and L. R. Petzold. Numerical solution of initial-value problems in differential-algebraic equations. SIAM, Philadelphia, 1996. Jing-Ru Cheng, Robert O. Strobl, Gour-Tsyh Yeh, Hsin-Chi Lin, and Woo Hee Choi. Modeling of 2D density-dependent flow and transport in the subsurface. J. Hydrologic Eng., 3(4):248--257, 1998. doi:10.1061/(ASCE)1084-0699(1998)3:4(248) A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker, and C. S. Woodward. SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers. ACM Transactions on Mathematical Software, 31(3):363--396, 2005. https://computation.llnl.gov/casc/nsde/pubs/toms_sundials.pdf P. S. Huyakorn, P. F. Andersen, J. W. Mercer, and Jr. H. O. White. Saltwater intrusion in aquifers: Development and testing of a three-dimensional, finite-element model. Water Resources Research, 23(2):293--312, 1987. F. Liu, V. V. Anh, I. Turner, K. Bajracharya, W. J. Huxley, and N. Su. A finite volume simulation model for saturated-unsaturated flow and application to Gooburrum, Bundaberg, Queensland, Australia. Applied Mathematical Modelling, 30(4):352--366, 2006. doi:10.1016/j.apm.2005.05.007 M. D. Tocci, C. T. Kelley, and C. T. Miller. Accurate and economical solution of the pressure-head form of richards' equation by the method of lines. Advances In Water Resources, 20(1):1--14, 1997. doi:10.1016/S0309-1708(96)00008-5 Q. Zhang, R. E. Volker, and D. A. Lockington. Numerical investigation of seawater intrusion at Gooburrum, Bundaberg, Queensland, Australia. Hydrogeology Journal, 12:674--687, 2004. doi:10.1007/s10040-004-0333-5

An accurate and reliable solution procedure for the EOQ model with deteriorating items under supplier credits linked to the ordering quantity

The inventory problem consists of two parts: (1) the modelling and (2) the solution procedure. The modelling can provide insight to solve the inventory problem and the solution procedure involves the implementation of the inventory model. Basically, the modelling and the solution procedure to the inventory problem are equally important. Chang et al.’s inventory model [Chang, C. T., Ouyang, L. Y., & Teng, J. T (2003). An EOQ model with deteriorating items under inflation when supplier credits linked to order quantity. Applied Mathematical Modelling, 27, 983–996] is correct and interesting. However, they ignore the explorations of the functional behaviors of the annual total relevant cost to locate the optimal solutions such that proofs of their solution procedures are not perfect from the viewpoint of logic. The main purposes of this paper are to provide accurate and reliable solution procedures to improve [Chang, C. T., Ouyang, L. Y., & Teng, J. T (2003). An EOQ model with deteriorating items under inflation when supplier credits linked to order quantity. Applied Mathematical Modelling, 27, 983–996].

Modeling and analysis of biodegradation of xenobiotic polymers based on experimental results

An endogenous depolymerization model based on uniform weight distribution is introduced. The time dependent model with temporal dependent degradation rate is reduced to a time independent model. The previously developed techniques were applied to an inverse problem to determine the degradation rate. The transition of weight distribution was simulated by solving an initial value problem. Those techniques are applied to degradation processes of polylactic acid. They are applicable to other polymers subject to endogenous depolymerization processes. References F. Kawai, Sphingomonads involved in the biodegradation of xenobiotic polymers, Journal of Industrial Microbiology and Biotechnology 23 (1999), 400--407. F. Kawai, M. Shibata, S. Yokoyama, S. Maeda, K. Tada, and S. Hayashi, Biodegradability of Scott--Gelead photodegradable polyethylene and polyethylene wax by microorganisms, Macromolecular Symposia 144 (1999), 73--84. F. Kawai, M. Watanabe, M. Shibata, S. Yokoyama, and Y. Sudate, Experimental analysis and numerical simulation for biodegradability of polyethylene, Polymer Degradation and Stability 76 (2002), 129--135. doi:10.1016/S0141-3910(02)00006-X F. Kawai, M. Watanabe, M. Shibata, S. Yokoyama, Y. Sudate, and S. Hayashi, Comparative Study on Biodegradability of Polyethylene Wax by Bacteria and Fungi, Polymer Degradation and Stability 86 (2004), 105--114. doi:10.1016/j.polymdegradstab.2004.03.015 J. D. Lambert, Computational Methods in Ordinary Differential Equations (John Wiley and Sons, Chichester, 1973). G. Madras and B. J. McCoy, Numerical and similarity solutions for reversible population balance equations with size-dependent rates, Journal of Colloid and Interface Science 246 (2002) 356--365. doi:10.1006/jcis.2001.8073 S. Matsumura, N. Tomizawa, A. Toki, K. Nishikawa and K. Toshima, Novel Poly(vinyl alcohol)-degrading enzyme and the degradation mechanism, Macromolecules 32 (1999), 7753--7761. B. J. McCoy and G. Madras, Evolution to similarity solutions for fragmentation and aggregation, Journal of Colloid and Interface Science 201 (1998) 200--209. doi:10.1006/jcis.1998.5434 B. J. McCoy, Distribution Kinetics for Temperature-Programmed Pyrolysis, Industrial and Engineering Chemistry Research 38 (1999) 4531--4537. B. J. McCoy and G. Madras, Discrete and continuous models for polymerization and depolymerization, Chemical Engineering Science 56 (2001) 2831--2836. doi:10.1016/S0009-2509(00)00516-9 J. E. J. Staggs, A continuous model for vaporization of linear polymers by random scission and recombination, Fire Safety Journal 40 (2005) 610--627. doi:10.1016/j.firesaf.2005.05.004 M. Watanabe and F. Kawai, Numerical simulation for enzymatic degradation of poly (vinyl alcohol), Polymer Degradation and Stability 81 (2003), 393--399. doi:10.1016/S0141-3910(03)00122-8 M. Watanabe and F. Kawai, Analysis of polymeric biodegradability based on experimental results and numerical simulation, Environmental Research and Control (in Japanese) 25 (2003), 25--32. M. Watanabe, F. Kawai, M. Shibata, S. Yokoyama, and Y. Sudate, Computational method for analysis of polyethylene biodegradation, Journal of Computational and Applied Mathematics 161 (2003), 133--144. doi:10.1016/S0377-0427(03)00551-X M. Watanabe, F. Kawai, M. Shibata, S. Yokoyama, Y. Sudate, and S. Hayashi, Analytical and computational techniques for exogenous depolymerization of xenobiotic polymers, Mathematical Biosciences 192 (2004) 19--37. doi:10.1016/j.mbs.2004.06.006 M. Watanabe and F. Kawai, Numerical simulation of microbial depolymerization process of exogenous type, R. May and A. J. Roberts (Eds.) Proc. of 12th Computational Techniques and Applications Conference, CTAC-2004, ANZIAM Journal, 46(E), pp. C1188--C1204, 2005. http://anziamj.austms.org.au/V46/CTAC2004/Wata M. Watanabe and F. Kawai, Mathematical modelling and computational analysis of enzymatic degradation of xenobiotic polymers, Applied Mathematical Modelling 30 (2006) 1497--1514. M. Watanabe, F. Kawai, S. Tsuboi, S Nakatsu, and H. Ohara, Study on enzymatic hydrolysis of polylactic acid by endogenous depolymerizaion model, Macromolecular Theory and Simulations 16 (2007) 619--626. doi:10.1002/mats.200700015 M. Watanabe and F. Kawai, Mathematical study of the biodegradation of xenobiotic polymers with experimental data introduced into analysis, Proceedings of the 7th Biennial Engineering Mathematics and Applications Conference, EMAC-2005, Melbourne, Editors: Andrew Stacey and Bill Blyth and John Shepherd and A. J. Roberts, ANZIAM J. 47 pp. C665--C681, 2007. http://anziamj.austms.org.au/V47EMAC2005/Watanabe

A COMPARISON OF HIGHER-ORDER COMPACT FINITE DIFFERENCE SCHEMES THROUGH BURGERS' EQUATION

Two higher-order compact finite difference approaches, the Hermitian and the Lax-Wendroff are examined by applying them to the viscous Burgers' equation. The difference equations obtained by the two methods were integrated in time through the third-order Strong-Stability-Preserving Runge-Kutta scheme. Absolute errors are computed by using an exact solution. The results are also compared with a second order central difference scheme. The Hermitian approach is far easier to implement. On uniform grids the Lax-Wendroff scheme produces smaller errors during the initial stages, but both methods are equally good for larger durations of integration. The convergence rate of the Hermitian scheme is slightly higher than the Lax-Wendroff scheme although both are of fourth order. Both schemes are unstable beyond a certain step size in time and space. When numerical boundary conditions are imposed, second-order conditions produce the best results whereas linear extrapolation proved to be the worst. It was also observed that large domains were required to implement the numerical boundary conditions properly. There was no detrimental effect on the accuracy of the results obtained through either of the two schemes when the size of the domain was greatly increased. Both schemes showed remarkable improvement in accuracy when clustered grids were employed. However much smaller time steps are required for stable solutions.

Wavefronts and global stability in a time-delayed population model with stage structure

We formulate and study a one–dimensional single–species diffusive–delay population model. The time delay is the time taken from birth to maturity. Without diffusion, the delay differential model extends the well–known logistic differential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes. This delayed model is known to have simple global dynamics similar to that of the logistic equation. Through the use of a sub/supersolution pair method, we show that the diffusive delay model continues to generate simple global dynamics. This has the important biological implication that quadratically regulated death processes dramatically simplify the growth dynamics. We also consider the possibility of travelling wavefront solutions of the scalar equation for the mature population, connecting the zero solution of that equation with the positive steady state. Our main finding here is that our fronts appear to be all monotone, regardless of the size of the delay. This is in sharp contrast to the frequently reported findings that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.

Crack blunting and the strength of soft elastic solids

When a material is so soft that the cohesive strength (or adhesive strength, in the case of interfacial fracture) exceeds the elastic modulus of the material, we show that a crack will blunt instead of propagating. Large–deformation finite–element model (FEM) simulations of crack initiation, in which the debonding processes are quantified using a cohesive zone model, are used to support this hypothesis. An approximate analytic solution, which agrees well with the FEM simulation, gives additional insight into the blunting process. The consequence of this result on the strength of soft, rubbery materials is the main topic of this paper. We propose two mechanisms by which crack growth can occur in such blunted regions. We have also performed experiments on two different elastomers to demonstrate elastic blunting. In one system, we present some details on a void growth mechanism for ultimate failure, post–blunting. Finally, we demonstrate how crack blunting can shed light on some long–standing problems in the area of adhesion and fracture of elastomers.

Limit analysis of ductile composites based on homogenization theory

A numerical method is presented in this paper for determining the load–bearing capacities of ductile composites such as metal matrix composites based on homogenization theory and the kinematic limit theorem. A representative volume element is chosen to reflect the microstructure of a periodic composite. By directly introducing the von Mises yield criterion into the kinematic limit theorem, a nonlinear optimization formulation can be obtained to calculate the ultimate strength of a ductile composite. The finite–element modelling of the kinematic limit analysis is formulated as a nonlinear mathematical programming problem with equality–constraint conditions, which can be solved by a direct iterative algorithm. An interface failure model based on the microscopic fluctuation displacements is proposed to account for the effects of interfaces on the failure of composites. The present method can effectively reveal the effects of the microstructure on the macroscopic properties and micromechanical failure mechanisms of composites. Finally, some numerical examples illustrate the validity of the present method.

A note on Demiray’s solution to the perturbed Burgers’ equation

The travelling wave ‘solution’ to the perturbed Burgers’ equation given by Demiray [Applied Mathematical Modelling 26 (2002) 37] is incorrect; an explanation as to why this is so is given.

A BEM FORMULATION FOR TRANSIENT AXISYMMETRIC THERMOELASTIC ANALYSIS USING PARTICULAR INTEGRALS

International Journal of Computational Engineering ScienceVol. 03, No. 04, pp. 339-353 (2002) No AccessA BEM FORMULATION FOR TRANSIENT AXISYMMETRIC THERMOELASTIC ANALYSIS USING PARTICULAR INTEGRALSD. P. HENRY, Jr., K. H. PARK, and P. K. BANERJEED. P. HENRY, Jr.Department of Civil Engineering, State University of New York at Buffalo, NY 14260, USA Search for more papers by this author , K. H. PARKSchool of Civil Engineering, Asian Institute of Technology, Pathumthani 12120, Thailand Search for more papers by this author , and P. K. BANERJEEDepartment of Civil Engineering, State University of New York at Buffalo, NY 14260, USA Search for more papers by this author https://doi.org/10.1142/S1465876302000678Cited by:3 Next AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AbstractA particular integral formulation is presented for the first time in a purely axisymmetric transient uncoupled thermoelastic analysis. The governing axisymmetric elastostatic equation is used as the complementary solution and the particular integrals for displacements and tractions are derived by integrating the three-dimensional formulation along the circumferential direction leading to elliptic integrals. The numerical results for three example problems are given and compared with their analytical solutions. Generally, the agreement among all of those results is satisfactory, if a few interior points are used to include the known values of temperature.Keywords:Boundary Element method (BEM)Particular IntegralsComplementary SolutionThermoelasticity FiguresReferencesRelatedDetailsCited By 3A cell-less BEM formulation for axisymmetric elastoplasticity via particular integralsAdisorn Owatsiriwong, Bupavech Phansri, Jung-Sik Kong and Kyung-Ho Park3 February 2009 | Computational Mechanics, Vol. 44, No. 2A new BEM formulation for transient axisymmetric poroelasticity via particular integralsK.H. Park and P.K. Banerjee1 Nov 2007 | International Journal of Solids and Structures, Vol. 44, No. 22-23A BEM formulation for inhomogeneous potential problems by particular integralsK.H Park1 Apr 2003 | Applied Mathematical Modelling, Vol. 27, No. 4 Recommended Vol. 03, No. 04 Metrics History KeywordsBoundary Element method (BEM)Particular IntegralsComplementary SolutionThermoelasticityPDF download

Yoffe–type moving crack in a functionally graded piezoelectric material

In this paper, the problem of a functionally graded piezoelectric material (FGPM) with a constantvelocity Yoffetype moving crack is considered. The strip is assumed to be under an antiplane mechani...

HIV-1 dynamics revisited: biphasic decay by cytotoxic T lymphocyte killing?

The biphasic decay of blood viraemia in patients being treated for human immunodeficiency virus type 1 (HIV-1) infection has been explained as the decay of two distinct populations of cells: the rapid death of productively infected cells followed by the much slower elimination of a second population the identity of which remains unknown. Here we advance an alternative explanation based on the immune response against a single population of infected cells. We show that the biphasic decay can be explained simply, without invoking multiple compartments: viral load falls quickly while cytotoxic T lymphocytes (CTL) are still abundant, and more slowly as CTL disappear. We propose a method to test this idea, and develop a framework that is readily applicable to treatment of other infections.

Generalization of the continuous and discrete structural approaches to the solution of problems of the dynamics of laminated composite plates and structures with stratification. Part 1. Applied mathematical model of the dynamic deformation of laminated composite structures

We construct an applied mathematical model of dynamic deformation of laminated structures. The proposed model can be regarded as a generalization of continous and discrete structural approaches to the analysis of laminated structures. This model enables one either to consider a structure as a single layer with reduced stiffness characteristics or split it across its thickness into a set of components each of which may include several layers and/or parts of layers. It is also possible to simulate various conditions of contact of layers (rigid or sliding). The differential operators appearing in the proposed model are similar to the operators used in the classical theory of shells, which simplifies the numerical realization of the model.

Integral equation methods in plane-strain elasticity with boundary reinforcement

In this paper, a linear theory of elastic boundary reinforcement of an elastic solid is developed for plane–strain deformations. The reinforcement consists of a thin elastic coating bonded to part of the boundary of the solid. The elastic properties of the coating incorporate both extensibility and bending rigidity. Interior and exterior mixed–boundary problems are formulated and solved using integral equation methods. The boundary value problems are reduced to systems of singular integro–differential equations to which Noether–type theorems are shown to apply. The case corresponding to a coating which has only extensibility properties and no bending rigidity is particularly interesting from a mathematical point of view and is given special attention. Finally, existence and uniqueness results are presented for both interior and exterior reinforcement problems of plane–strain.

The writhing of circular cross–section rods: undersea cables to DNA supercoils

The large deflection theory circular cross–section elastic rods is used to consider the writhing of long straight rods subjected to tension and torque, such as undersea cables, and to closed loops with inserted twist, such as DNA supercoils. The writhed shape of the long straight rod under tension and torque is easily generated by twisting a piece of string with the fingers and consists of three separate parts: a balanced ply region, a free end loop, and two tail regions. The solution for the rod shape in each of the regions is found. The results are then joined together to ensure continuity of the position and tangent vectors of the strand centre line through the introduction of point forces and moments at the points where the strands enter and exit the balanced ply. The results of the model are consistent with simple experiments on long braided rope. The writhed shape of the closed loop with twist inserted between the ends prior to closure is modelled as a balanced ply joined to two end loops. The analysis combines the mechanics solution with the conservation of topological link to provide a simple formula which quantitatively predicts the approximate shape and helix angle of the supercoil. The results are in good agreement with simple experiments on rope and with available data on DNA supercoils.

Enhancing the accuracy of the solution to unsaturated flow by a Hermitian Green element model

Our earlier paper demonstrated that the singular boundary element (BE) theory, which hitherto had been considered unsuitable for the highly nonlinear variably saturated transient flow problem, could be applied to the same problem when implemented along the lines of the Green element method (GEM) (Taigbenu, A. E. and Onyejekwe, O. O., Green element simulations of transient nonlinear unsaturated flow equation, Applied Mathematical Modelling, 1995, 19, 675–684). Here, the transient one-dimensional unsaturated flow problem is revisited with a Green element model which incorporates the cubic Hermitian interpolation basis functions to approximate the distribution of the primary variable and the soil constitutive relations. Because the soil parameters vary appreciably by several orders of magnitude over small intervals of soil moisture, approximating those parameters by linear interpolation functions, as earlier done, could be inadequate, and this fact is demonstrated here using two numerical examples of infiltration into vertical soil columns. With the first example, there is good agreement between the solutions of Hermitian GEM and Hermitian finite element method (FEM) which confirms that the discrepancy of less amount of soil moisture in the linear GEM solution, earlier observed, is due to an error from interpolation (Taigbenu, A. E. and Onyejekwe, O. O., Green element simulations of transient nonlinear unsaturated flow equation, Applied Mathematical Modelling, 1995, 19, 675–684). The second example serves to show that comparable accuracy between Hermitian and linear GE models can be achieved using a fewer number of elements in the former model.

A unified theory for the interpretation of total pressure and temperature in two-phase flows at subsonic and supersonic speeds

This paper presents a unified theory on the interpretation of total pressure and total temperature in multiphase flows. The present approach applies to both vapour-droplet mixtures and solid-particle laden gases, and at subsonic as well as supersonic velocities. It is shown here that the non-equilibrium processes occurring in the vicinity of a stagnation point are important. These processes may be responsible for the generation of entropy and affect the pressure and temperature at the stagnation point. They should be properly considered while inferring, say, flow velocity or entropy generation from Pitot measurements. By proper non-dimensionalization of the relevant parameters, it is possible to find a single (theoretically obtained) calibration curve for the total pressure as a function of the particle size, which is almost independent of the constituents of the multiphase mixture and of the flow conditions. The calibration curve is a plot of a pressure recovery factor versus Stokes number and specifies the total pressure under different non-equilibrium conditions. The total pressure, predicted by the present theory, varies monotonically between the two limiting values: the frozen total pressure (when there is no interphase mass, momentum and energy transfer in the decelerating flow towards the stagnation point) and the equilibrium total pressure (when the dispersed phase, either the liquid droplets or the solid particles, is always at inertial and thermodynamic equilibrium with the continuous vapour phase). The equilibrium total pressure is always higher than the frozen total pressure. It is shown that the equilibrium total temperature, on the other hand, may be higher or lower than the frozen total temperature. In addition, unlike the case of total pressure, the calibration curve for total temperature is not so universal, and the total temperature under non-equilibrium conditions is not necessarily bounded between the frozen and equilibrium values. It is further shown that the entropy of a multiphase mixture has to be carefully interpreted and is not unequivocally related to the total pressure even in steady adiabatic flow.

Yarn twist in the ring-spinning balloon

A dynamical theory of twist insertion in ring-spun yarns that brings together the previous work of the authors and their colleagues on the stability of the ring-spinning balloon, and the recent wor...

Progress toward a mining strategy based on mineral leaching with secondary recovery : L. K. Forbes, Applied Mathematical Modelling, 20(1), 1996, pp 16–25

Progress toward a mining strategy based on mineral leaching with secondary recovery : L. K. Forbes, Applied Mathematical Modelling, 20(1), 1996, pp 16–25

A level II reliability approach to tunnel support design: E. Laso, M. S. Gomez Lera & E. Alarcon, Applied Mathematical Modelling, 19(6), 1995, pp 371–382

A level II reliability approach to tunnel support design: E. Laso, M. S. Gomez Lera & E. Alarcon, Applied Mathematical Modelling, 19(6), 1995, pp 371–382

A dynamic model of ground water restoration with accelerator constraint: F. Firooi & J. Merrifield, Applied Mathematical Modelling, 19(3), 1995, pp 173–181

A dynamic model of ground water restoration with accelerator constraint: F. Firooi & J. Merrifield, Applied Mathematical Modelling, 19(3), 1995, pp 173–181